A STUDY OF SINGLE-LAP JOINTS
By
Matthew Paul Lempke

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering – Master of Science
2013

ABSTRACT

A STUDY OF SINGLE-LAP JOINTS
By
Matthew Paul Lempke

Single-lap joints are a widely-used and relatively strong and simple way joining two
materials via an overlapping bond. With the growing use of composite materials in
modern design practices, the need to join increasingly dissimilar materials has arisen. As
such, knowledge concerning the behavior of single-lap joints with dissimilar adherends is
essential.

To investigate the behavior of single-lap joints as material and geometric properties are
varied under tensile loading, an analytically verified finite element parametric study was
conducted on both ideally and adhesively bonded single-lap joints, measuring the
changes in stress value at points of critical stress concentrations. In order to correlate the
finite element analysis with real-world lap joint behavior, digital image correlation was
used to record the deformation of lap joint specimens under a tensile load. A finite
element model was then developed, and compared, to the experimental results.

With the results of the parametric study and experimental comparison, trends in stress
changes were identified and explained, and design suggestions were made based on these
trends. The results of the experimental finite element model were reasonably correlated,
and several suggestions for improvements were made.

ACKNOWLEDGEMENTS

I would like to thank Dr. Dahsin Liu for allowing me the opportunity and guiding me
through the research and thesis writing process. I would also like to thank the US Army
Research Laboratory for funding this research, and Dr. Mahmood Haq and Dr. Xinran
Xiao for serving on my thesis defense committee, and helping me finalize my thesis.
Lastly, I would like to thank my uncle William Janis for always believing in me and
making a college education possible, Alexis Palmquist, for always standing with me and
being there whenever I needed her, and the rest of my friends and family for being the
best support structure anyone could want.

iii

TABLE OF CONTENTS
LIST OF TABLES ............................................................................................................ v
LIST OF FIGURES ........................................................................................................ vii
KEY TO SYMBOLS ........................................................................................................ x
1 Introduction .................................................................................................................... 1
REFERENCES.............................................................................................................. 5
2 Brief Overview of the Single-lap Joint ......................................................................... 7
REFERENCES............................................................................................................ 10
3 Theoretical Analysis .................................................................................................... 12
REFERENCES............................................................................................................ 18
4 Finite element analysis of the Single-Lap Joint......................................................... 20
REFERENCES............................................................................................................ 35
5 Results of the FEA Based Parametric Study ............................................................. 37
5.1 Varied Adherend Thickness Under Ideal Bonding and a Tensile Load .......... 37
5.2 Varied Adherend Thickness Under Ideal Bonding and a Tensile Load .......... 43
5.3 Varied Free Adherend Length Under Ideal Bonding and a Tensile Load ...... 47
5.4 Varied Bonding Length Under Ideal Bonding and a Tensile Load ................. 51
5.5 Varied Young’s Modulus Under Ideal Bonding and a Three-Point Bending
Load.............................................................................................................................. 55
5.6 Varied Adherend Thickness Under Ideal Bonding and a Three-Point-Bending
Load.............................................................................................................................. 59
5.7 Varied Free Adherend Length Under Ideal Bonding and a Three-PointBending Load .............................................................................................................. 63
5.8 Varied Bonding Length Under Ideal Bonding and a Three-Point-Bending
Load.............................................................................................................................. 66
5.9 Varied Adhesive Thickness Under Adhesive Bonding and a Tensile Load..... 69
5.10 Varied Young’s Modulus Under Adhesive Bonding and a Tensile Load ...... 73
5.11 Varied Adherend Thickness Under Adhesive Bonding and a Tensile Load . 78
5.12 Varied Adhesive Young’s Modulus Under Adhesive Bonding and a Tensile
Load.............................................................................................................................. 81
5.13 Design Recommendations via Parametric Study ............................................. 84
6 Experimental Validation ............................................................................................. 86
REFERENCES.......................................................................................................... 101
7 Conclusions ................................................................................................................. 103
8 Suggestions for Future Research .............................................................................. 107
iv

LIST OF TABLES

Table 4.1 Tensile basic case stress values at points A and B ....................................... 25
Table 4.2 Tensile basic case stress values at points A and B ....................................... 30
Table 5.1 Reference stress values and SCFs at point A with varying E/E0 ............... 38
Table 5.2 Reference stress values and SCFs at point B with varying E/E0 ............... 38
Table 5.3 Reference stress values and SCFs at point A with varying t/t0 .................. 43
Table 5.4 Reference stress values and SCFs at point B with varying t/t0 .................. 44
Table 5.5 Reference stress values and SCFs at point A with varying L/L0 ............... 47
Table 5.6 Reference stress values and SCFs at point B with varying L/L0 ............... 48
Table 5.7 Reference stress values and SCFs at point A with varying c/c0 ................. 51
Table 5.8 Reference stress values and SCFs at point B with varying c/c0 ................. 52
Table 5.9 Reference stress values and SCFs at point A with varying E/E0 ............... 55
Table 5.10 Reference stress values and SCFs at point B with varying E/E0 ............. 56
Table 5.11 Reference stress values and SCFs at point A with varying t/t0 ................ 59
Table 5.12 Reference stress values and SCFs at point B with varying t/t0 ................ 60
Table 5.13 Reference stress values and SCFs at point A with varying L/L0 ............. 63
Table 5.14 Reference stress values and SCFs at point B with varying L/L0 ............. 64
v

Table 5.15 Reference stress values and SCFs at point A with varying c/c0 ............... 66
Table 5.16 Reference stress values and SCFs at point B with varying c/c0 ............... 67
Table 5.17 Stress values and SCFs as η/η0 varies at point A1 .................................... 70
Table 5.18 Stress values and SCFs as η/η0 varies at point B2..................................... 70
Table 5.19 Stress values of the basic model without any adhesive ............................. 73
Table 5.20 Stress values of the basic model with adhesive, SCFs compared to table 1
.......................................................................................................................................... 74
Table 5.21 Stresses and SCFS at point A1 as E/E0 varies ........................................... 75
Table 5.22 Stresses and SCFS at point B2 as E/E0 varies ........................................... 75
Table 5.23 Stresses and SCFS at point A1 as t/t0 varies .............................................. 78
Table 5.24 Stresses and SCFS at point B2 as t/t0 varies .............................................. 78
Table 5.25 Stresses and SCFS at point A1 as Ec/Ec0 varies ........................................ 81
Table 5.26 Stresses and SCFS at point B2 as Ec/Ec0 varies ........................................ 81
Table 6.1 Dimensions of specimen 1 .............................................................................. 88
Table 6.2 Dimensions of specimen 2 .............................................................................. 89
Table 6.3 Comparison of displacements for specimen 1 ............................................. 91
Table 6.4 Comparison of displacements for specimen 2 ............................................. 92
Table 6.5 Failure stress predictions for specimen 1 ................................................... 100
Table 6.6 Failure stress predictions for specimen 2 ................................................... 100

vi

LIST OF FIGURES

Figure 2.1 A single-lap joint ............................................................................................. 7
Figure 2.2 A double-lap joint ........................................................................................... 8
Figure 2.3 A step-lap joint ................................................................................................ 8
Figure 3.1 Analytical solution geometry and boundary conditions ........................... 13
Figure 3.2 Analytical solution ........................................................................................ 15
Figure 3.3 Analytical solution and FE solution ............................................................ 17
Figure 4.1 Analytical solution and FE solution ............................................................ 22
Figure 4.2 Finite element boundary conditions in tension .......................................... 23
Figure 4.3 Finite element boundary conditions in three point bending ..................... 23
Figure 4.4 Von-Mises stress contour map under unit tensile loading ........................ 26
Figure 4.5 X direction stress contour map under unit tensile loading ....................... 27
Figure 4.6 Y direction stress contour map under unit tensile loading ....................... 28
Figure 4.7 XY shear stress contour map under unit tensile loading .......................... 29
Figure 4.8 Von-Mises stress contour map under three point bending load .............. 31
Figure 4.9 X direction stress contour map under three point bending load ............. 32
Figure 4.10 Y direction stress contour map under three point bending load ........... 33
Figure 4.11 XY shear stress contour map under three point bending load .............. 34
Figure 5.1 Ideally bonded Young’s Modulus variation critical points ...................... 38
Figure 5.2 SCFs as E/E0 varies at point A .................................................................... 39
Figure 5.3 SCFs as E/E0 varies at point B .................................................................... 40
Figure 5.4 Ideally bonded thickness variation critical points ..................................... 43
vii

Figure 5.5 SCFs as t/t0 varies at point A ....................................................................... 44
Figure 5.6 SCFs as t/t0 varies at point B ....................................................................... 45
Figure 5.7 Ideally bonded free adherend length variation critical points ................. 47
Figure 5.8 SCFs as L/L0 varies at point A ..................................................................... 48
Figure 5.9 SCFs as L/L0 varies at point B ..................................................................... 49
Figure 5.10 Ideally bonded bonding length variation critical points ......................... 51
Figure 5.11 SCFs as c/c0 varies at point A .................................................................... 52
Figure 5.12 SCFs as c/c0 varies at point B .................................................................... 53
Figure 5.13 Ideally bonded Young’s Modulus variation critical points .................... 55
Figure 5.14 SCFs as E/E0 varies at point A .................................................................. 56
Figure 5.15 SCFs as E/E0 varies at point B .................................................................. 57
Figure 5.16 Ideally bonded thickness variation critical points ................................... 59
Figure 5.17 SCFs as t/t0 varies at point A ..................................................................... 60
Figure 5.18 SCFs as t/t0 varies at point B ..................................................................... 61
Figure 5.19 Ideally bonded free adherend length variation critical points ............... 63
Figure 5.20 SCFs as L/L0 varies at point A ................................................................... 64
Figure 5.21 SCFs as L/L0 varies at point B ................................................................... 65
Figure 5.22 Ideally bonded bonding length variation critical points ......................... 66
Figure 5.23 SCFs as c/c0 varies at point A .................................................................... 67
Figure 5.24 SCFs as c/c0 varies at point B .................................................................... 68
Figure 5.25 Finite element boundary conditions and parameters .............................. 69
viii

Figure 5.26 Adhesively bonded adhesive thickness variation critical points ............ 70
Figure 5.27 SCFs as η/η0 varies at point A1 ................................................................. 71
Figure 5.28 SCFs as η/η0 varies at point B2 ................................................................. 72
Figure 5.29 Adhesively bonded Young’s Modulus variation critical points ............. 74
Figure 5.30 SCF at point A1 as E/E0 varies .................................................................. 76
Figure 5.31 SCF at point B2 as E/E0 varies .................................................................. 77
Figure 5.32 Adhesively bonded adherend thickness variation critical points ........... 78
Figure 5.33 SCF at point A1 as t/t0 varies ..................................................................... 79
Figure 5.34 SCF at point B2 as t/t0 varies ..................................................................... 80
Figure 5.35 Adhesively bonded adhesive Young’s Modulus variation critical points
........................................................................................................................................... 81
Figure 5.36 SCF at point A1 as Ec/Ec0 varies ............................................................... 82
Figure 5.37 SCF at point B2 as Ec/Ec0 varies ............................................................... 83
Figure 6.1 Diagram of the dimensions and shapes of the experimental specimens .. 87
Figure 6.2 Finite element (left) and DIC (right) displacement magnitude specimen 1
........................................................................................................................................... 92
Figure 6.3 Finite element (left) and DIC (right) X direction displacement specimen 1
........................................................................................................................................... 93
Figure 6.4 Finite element (left) and DIC (right) Y direction displacement specimen 1
........................................................................................................................................... 94
Figure 6.5 Finite element (left) and DIC (right) displacement magnitude specimen 2
........................................................................................................................................... 96
Figure 6.6 Finite element (left) and DIC (right) X direction displacement specimen 2
........................................................................................................................................... 97
Figure 6.7 Finite element (left) and DIC (right) Y direction displacement specimen 2
........................................................................................................................................... 98
ix

KEY TO SYMBOLS

c

Half bonding length

c0

Half base bonding length

E

Adherend Young’s Modulus

E0

Base Young’s Modulus

Ec

Adhesive Young’s Modulus

Ec0

Base adhesive Young’s Modulus

k

Edge moment factor

L

Free adherend length

L0

Base free adherend length

M0

Edge moment

P

Applied three-point bending load

T

Applied tensile load

t

Adherend thickness

t0

Base adherend thickness

w1

Adherend transverse axis

w2

Adhesive transverse axis

x1

Adherend normal axis

x2

Adhesive normal axis

αn

Eccentric loading angle

η

Adhesive thickness

σ0

Applied tensile pressure
x

σvm

Von-Mises stress

σxx

X direction stress

σyy

Y direction stress

τxy

Shear stress

ν

Adherend Poisson’s Ratio

νc

Adhesive Poisson’s Ratio

xi

1 Introduction

As composite materials become more prevalent in modern design, many situations arise
in which they are needed in conjunction with traditional homogeneous materials.
Namely, in vehicles and many other applications, metal to composite joining is necessary
to increase the strength to weight ratio in modern structures. However, some difficulties
arise in the joining of these dissimilar materials. Where one could presumably weld two
metals together, or mix ceramic powders into the complicated shape desired, one cannot
join polymeric composites to metal in such a strong manner. Other methods of joining are
possible, such as mechanical fastening, or adhesive joining. The focus of this thesis is to
investigate how the variation of dominant joint material and geometric parameters
independently affect the critical stresses in adhesive single-lap joints as predicted by
finite element analysis, and verify this analysis both analytically and experimentally.

Lap joints, specifically the adhesive single-lap joint, have been studied thoroughly
throughout the years. Analytic solutions date back as far as Volkerson [1] and his
simplified solution in 1938 that is still more accurate than the current ASTM standards
D1002-10 and D3983-98 used to determine shear strength and shear modulus,
respectively. Goland and Reissner [2] built upon his solution, adding the influence of the
eccentric loading angle and thus introducing the important influence of the edge moment
by adding the bending moment factor, k. This factor is related to the bending moment M0,
the adherend thickness t, and the load applied to the adherends T by equation 1.1.

1

k

2M 0
Tt

(1.1)

This bending moment factor is unity for small loadings for which no rotation of the joint
itself results, and less than unity for higher loads. They found that maximum stresses
occur at the ends of the overlap, which is to be expected. However, according to Adams
[3], Goland and Reissner’s solution is not applicable to most practical joints, or joints
including adherends of differing materials, as they neglect adherend shear strains He
found that a geometrically non-linear finite element analysis is applicable to a wider
range of lap joints. Tsai and Morton [4] performed an in depth comparison of the
available analytic solutions in 1992, and found Hart-Smith’s [5] model the most accurate
for determining the edge moment in short joints, and Oplinger’s [6] model the most valid
for determining the edge moment in long joints. Goland and Reissner’s solution predicted
a global sense of the adhesive stress distributions for both lengths most effectively, and
was altogether more accurate than the other solutions for adherends of intermediate
length. Tsai and Morton [4] defined long lap joints as those with a free adherend length to
bonding length ratio of greater than 5 and short joints as those with a ratio of less than
0.75. They also found that geometrically non-linear finite element analysis was still the
most effective and accurate method of single-lap joint modeling.

Carpenter [7] explored the effects of various mathematical assumptions on the adhesive
stresses in single-lap joints, and found the predicted maximum adhesive shear and peel
stresses were mostly unaffected by most assumptions, but neglecting shear deformation

2

of the adherends affects peel stress significantly. He again found that finite element
methods yield results very close to those from lap joint theories such as Goland and
Reissner [2] and Delale and Erdogan [13]. Cooper and Sawyer [8] further explored finite
element analysis, comparing geometrically linear results to non-linear results. The results
were then compared to Goland and Reissner’s analytical solution and photoelastic
experiments, and it was shown that the geometrically non-linear Finite Element Analysis
was accurate, and the non-linearity had a large effect on the stresses in the adhesive.
Goland and Reissner’s solution was found to be sufficient for the prediction of stresses
along the midline of the adhesive bond.

Lap joints have also been thoroughly experimentally and numerically analyzed in a
variety of forms and methods. Crocombe and Adams [9] explored the influence of a spew
fillet on the stress distributions in a single-lap joint, and found that maximum adhesive
stresses are usually much lower with the addition of the spew fillet rather than a square
termination of the adhesive. Spew fillets are triangular adhesive fillets at the termination
of the bond. More recently, Shin, Lee, and Lee [10] studied shear strength of co-cured
single-lap joints subjected to tensile loads, and found that in most cases, the failure
mechanism was partially cohesive failure, and the shear strength is significantly impacted
by bond length and stacking sequence. Elaborating, the stacking sequence of the
composite adherend greatly affects the shear strength of the joint due to the fact that the
stacking sequence determines the difference in stiffness between adherends. Also, as
bond length increases, so does the shear strength of the lap joint. Maglahaes, de Moura,
and Gonclaves [11] explored the stress concentration effects in laminate composite

3

single-lap joints through two-dimensional Finite Element Analysis, and found critical
stress locations near the ends of the overlap and discussed their role in damage initiation
in composite lap joints. Grant, Adams, and de Silva [12] investigated experimental and
numerical analysis of single-lap joints for the automotive industry, including tension,
four-point, and three-point bending tests varying other parameters as well. They found
three-point bending and tension tests yielded similar results in the adhesive, as they both
induce a large bending moment and both initiate failure in the adhesive, while four-point
bending did not cause any adhesive failure. The adherend material (steel) yielded before
the adhesive, and they proposed a failure criterion.

This work will first define lap joints and present an overview of several of the various lap
joint geometries. Analytical derivation and verification is then presented, followed by a
description and in-depth example of the parametric study conducted. The results are then
presented, and discussed, followed by experimental work and validation, and both a
conclusions and suggestions section.

4

REFERENCES

5

REFERENCES

[1] Volkerson, O. “Die Niektraftverteiling in Zugbeanspruchten mit Konstanten
Laschquershritten.” Luftfahrforschung. 48. (1938)
[2] Goland, M., Reissner, E. “The stresses in cemented joints.” J. Appl. Mech. 11. (1944)
[3] Adams, R. D. “Strength Predictions for Lap Joints, Especially with Composite
Adherends. A Review.” J. of Adhesion. 30. (1989)
[4] Tsai, M. Y., Morton, J. “An Evaluation of Analytical and Numerical Solutions to the
Single-Lap Joint.” Int. J. Solids Structures. 31. (1994)
[5] Hart–Smith, L. J. “Adhesive-bonded single-lap joints.” NASA. CR-112236. (1973)
[6] Oplinger, D. W. “A layered beam theory for single lap joints.” Army Materials
Technology Laboratory Report. MTL TR91-23. (1991)
[7] Carpenter, W. “A comparison of numerous lap joint theories for adhesively bonded
joints” J. of Adhesion. 35. (1991)
[8] Cooper, P. A., Sawyer, J. W. “A Critical Examination of Stresses in an Elastic Single
Lap Joint.” NASA. TR1507. (1979)
[9] Crocombe, A. D., Adams, R. D. “Influence of the Spew Fillet and other Parameters
on the Stress Distribution in the Single-Lap Joint.” J. of Adhesion. 13. (1981)
[10] Shin, K. C., Lee, J. J., Lee, D. G. “A study on the lap shear strength of a co-cured
single lap joint.” J. of Adhesion Science and Technology. 14. (2012)
[11] Maglahaes, A. G., de Moura, M. F. S. F., Gonclaves, J. P. M. “Evaluation of stress
concentration effects in single-lap bonded joints of laminate composite materials.”
International Journal of Adhesion and Adhesives. 25. (2005)
[12] Grant, L. D. R., Adams, R. D., de Silva, L. F. M. “Experimental and numerical
analysis of single-lap joints for the automotive industry.” International Journal of
Adhesion and Adhesive. 29. (2009)
[13] F. Delale, F. Erdogan. “Stresses in Adhesively Bonded Joints: A Closed-Form
Solution.” Journal of Composite Materials. 15. (1981)

6

2 Brief Overview of the Single-lap Joint

A full single-lap joint is simply an anti-symmetric structure of two materials, known as
adherends, bonded via an overlap, usually with adhesive, bolts, or both, where no
material is removed at the bond. In other words, a full lap is like Figure 2.1, a simple
single-lap joint. There is a full overlap with no adherend material altered at the joint.

Figure 2.1 A single-lap joint

Half lap joints can have material removed at the bond for symmetry, or to keep the
surface smooth and without overlap. Double-lap joints are a full lap joint that simply has
an additional adherend, as illustrated in Figure 2.2. A step lap is a half lap joint with a
step-like interface, shown in Figure 2.3. There are many types of lap joints, such as but
in this thesis, only ideally bonded, a theoretical ideal in which no adhesive is used, and
adhesively bonded single-lap joints with square adhesive termination are investigated.
Square adhesive termination refers to the fact that there is no fillet at the edge of the
adhesive. Fillets have been shown to increase failure stress in lap joints, as shown in the
literature survey in chapter 1.

7

Figure 2.2 A double-lap joint

Figure 2.3 A step-lap joint

Several parameters influence the stress distributions and strength of single-lap joints.
Important material properties include the various moduli and Poisson’s Ratios of each
adherend and the adhesive. Geometrically, both the free length and bond length of each
adherend, their thickness and the thickness of the adhesive also impact the stress
distributions and overall displacement of the structure. The effects of these parameters is
investigated and explained in chapter 5.

Dissimilar joining, or joining two adherends of differing material property or geometry, is
difficult. For instance, joining a composite adherend to a metal adherend presents several
issues, as the adhesive may not bond to the dissimilar surfaces equally as well, and any
holes drilled in the composite for mechanical fastening creates stress concentrations that
8

may initiate damage in the composite due to its non-homogeneous Young’s Modulus.
The differing properties between the two materials could influence the stress distributions
to be more intense than a homogeneous single-lap joint. Therefore, this thesis investigates
the important parameters’ impact on the critical stress points in the adherends of singlelap joints.

9

REFERENCES

10

REFERENCES

[1] Budynas, Richard G., and Nisbett, Kieth J. Shigley’s Mechanical Engineering Design.
9th ed. New York: McGraw Hill, 2011. Print.

11

3 Theoretical Analysis

In order to confirm the validity and accuracy of the finite element analysis presented later
in this work, it was necessary to explore analytical solutions for the displacements in a
single lap joint. While several were presented, Goland and Reissner’s [1] equations were
found to be the most accurate for the parametric study conducted, and others, like Tsai
and Morton [2], used this solution to validate their finite element analysis.
The several other solutions mentioned include Volkerson’s [3] famous equation presented
in 1938, which was an improvement on the ASTM standard for shear strength
determination (ASTM D1002-10) and modulus determination (ASTM D3983-98) for
adhesives, still used today. In Tsai and Morton’s review, Volkerson’s solution was found
to be inferior to the other solutions reviewed, and the solution itself can be found in his
publication. Goland and Reissner’s [1] solution was the first to include the influence of
the edge moment, the moment that arises at the edge of the overlap, which was an
important step towards an accurate single lap joint model. Goland and Reissner found
that this edge moment is the dominant factor in the stress development throughout the
joint geometry, and its inclusion made their solution a leap ahead of Volkerson’s. HartSmith [4] and Oplinger [5] also presented different models. Tsai and Morton found that
Hart-Smith’s model was the most accurate in predicting the stresses in short single-lap
joints, while Oplinger’s solution was found to be more accurate for long single-lap joints.
Long and short joint cases are defined by Tsai and Morton as joints in which the ratio of
l/c is greater than or equal to 10 or less than or equal to 1.25, respectfully. Joints falling
between these bounds are considered adhesive joints of intermediate length. However,

12

Goland and Reissner’s solution was still found to be accurate in its prediction of
deflection, and thus it is used for the validation of the following work.

Figure 3.1 Analytical solution geometry and boundary conditions
With reference to Figure 3.1, x1 being the region in the adherend bounded by zero and L,
and x2 being the region in the adhesive bounded by zero and 2c, Goland and Reissner
present the following general solution and boundary conditions:

d 2 w1
dx12



3
T
 n x1  w1   0, 0  x1  L, D1  Et 2
D1
12 1  



d 2 w2 T 
t

2
 n L  x2   w2  2   0, 0  x2  2c,
D2 
dx2


2 Et 3
D2 
3 1 2







(3.1)

(3.2)

Subject to simple support external boundary conditions and the internal boundary
conditions including the continuous displacements and rotation angles and zero
displacement at the anti-symmetric point:
w1 (0)  0, w2 (c)  0, w1 ( L)  w2 (0),

13

dw1
dw2
 x 2  0
( x1  L) 
dx1
dx2

(3.3)

The general solution to each equation is of the form:

w1  A1 cosh u1 x1   B1 sinh u1 x1    n x1 , 0  x1  L

t
w2  A2 cosh u 2 x 2   B2 sinh u 2 x 2    n 1  x 2 

2 n

T
and u 2 
D1

where u1 

(3.4)

, 0  x 2  2c



(3.5)
(3.6)

T
D2

Applying the boundary conditions and solving, the constants can then be found to be:

A1  0

(3.7)

B1 

(3.8)

 u 2 u1 cosh u1 L sinh u 2 c   u 2 cosh u 2 c sinh u1L 
4 n 1  c   2t  coshu 2 c 2t   n 4 L  4



t 
t
   n u 2 sinh u1 L  c  1 
u1 cosh u1 L  sinh u 2 c  n  L  1 



2 n 
2 n


A2 
u1 cosh u1 L  sinh u 2 c   u 2 cosh u 2 c  sinh u1 L 
B2 

 u1 cosh u1 L  n 2  2c   cosh u 2 c t   n 2 L  2  t
2u1 cosh u1 L  sinh u 2 c   u 2 cosh u 2 c  sinh u1 L 






(3.9)

(3.10)

Due to homogeneity of materials and boundary conditions, one can plot the region from –
L to 0, and 0 to 2c, obtaining the analytic solution plot in Figure 2, below.

14

Transverse Deflection
0.3
0.2

Deflection/t

0.1
0
-0.1
-0.2
-0.3
-0.4
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

x/L

Figure 3.2 Analytical solution

Of particular note in this analysis is the eccentric angle of loading, αn. Because of this, as
loading increases, non-linear geometric effects occur. Therefore, any finite element
analysis performed needs to include considerations for high nodal rotations and a
sufficiently small (5-10% of the total simulation time) timestep.

Performing a geometrically non-linear finite element analysis with ABAQUS and
constraining the adherends with the same boundary conditions as Goland and Reissner’s

15

problem statement for zero adhesive thickness, the appropriate displacements were
obtained, and Figure 3.3 was plotted, which compares Figure 3.2 and the finite element
transverse deflection results, measured along the centerline of the adherend as defined by
the x1, w1 coordinate system, and the centerline of the adhesive as defined by the x2, w2
coordinate system. The finite element analysis was performed on a typical short joint as
dictated by Tsai and Morton, and a load of T = 400 N was applied as shown in Figure
3.1. It is immediately obvious that ABAQUS accurately solves the model, with FE
displacements extremely similar to analytical displacements, using an adequately fine
mesh.

16

Transverse Deflection
0.3

0.2

Goland and Reissner
NLFEM

Deflection/t

0.1

0

-0.1

-0.2

-0.3

-0.4
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

x/L

Figure 3.3 Analytical solution and FE solution

17

0.4

REFERENCES

18

REFERENCES

[1] Volkerson, O. “Die Niektraftverteiling in Zugbeanspruchten mit Konstanten
Laschquershritten.” Luftfahrforschung. 48. (1938)
[2] Goland, M., Reissner, E. “The stresses in cemented joints.” J. Appl. Mech. 11. (1944)
[3] Tsai, M. Y., Morton, J. “An Evaluation of Analytical and Numerical Solutions to the
Single-Lap Joint.” Int. J. Solids Structures. 31. (1994)
[4] Hart–Smith, L. J. “Adhesive-bonded single-lap joints.” NASA. CR-112236 (1973)
[5] Oplinger, D. W. “A layered beam theory for single lap joints.” Army Materials
Technology Laboratory Report. MTL TR91-23. (1991)

19

4 Finite element analysis of the Single-Lap Joint

Due to the relative difficulty in obtaining analytical solutions for single-lap joints subject
to various loading conditions, many instead use finite element analysis to study their
stresses. For example, Tsai and Morton [1] evaluated analytical and finite element
analysis solutions to single-lap joints. While several solutions were investigated, as
presented in chapter three of this thesis, none were universally valid for all single-lap
joints. They found that Hart-Smith’s [2] solution was most accurate in predicting stresses
in short single-lap joints, and Oplinger’s [3] solution was the most accurate for long
single-lap joints. However, Finite element analysis with non-linear geometrical effects
included in the analysis is universally applicable to single-lap joints. Tsai and Morton
performed two-dimensional plane-strain finite element analysis with material linearity
and geometric nonlinearity in order to investigate the edge moment, analyze the state of
stress in the adhesive layer, and evaluate the analytical solutions of Goland and Reisner
[4], Hart-Smith [2], and Oplinger [3], reaching the conclusions listed above.

Grant, Adams, and de Silva [5] experimentally and numerically studied single-lap joints
in context of the automotive industry, and regarding finite element analysis, used a twodimensional plane-strain model with isoparametric elements. They found that the finite
element results were very sensitive to mesh size, as mathematical singularities exist at the
ends of the overlap at the adherend-adhesive interface. In order to combat this, they
simply conducted comparisons through similar meshes. Adams [6] utilized finite element
analysis in order to predict single-lap joint strength, and found that finite element analysis
was the most efficient method for analyzing single-lap joints with their geometric
20

nonlinearity and possible corner rounding or complicated adhesive termination, like spew
fillets. He commented that analytical techniques cannot be used to predict the strength of
adhesively-bonded lap joints without an uncertainty factor, as they cannot adequately
describe the real stress and strain conditions at the ends of the joint, where failure
initiates.

In order to adequately model single-lap joints and how their important parameters affect
the stresses in the adherends and adhesive, two-dimensional plane-strain analysis with a
consistent isoparametric mesh was conducted across all cases presented. [1] and [5] both
encountered the issue of stress singularity at the sharp corners of a single-lap joint, and
both combatted the issue with a consistent mesh; a mesh that measures stress from the
same distance from the singularity. In the case of this thesis, stress was always measured
at the same point across cases, and it was always measured at the centroid of the element,
with 40 elements per basic case adherend thickness. This allowed for a comparison of
stresses across cases, without danger of influence from the singularity. The good
correlation between analytical deflection and finite element deflection in Figure 4.1, represented below, verifies this mesh scheme.

21

Transverse Deflection
0.3

0.2

Goland and Reissner
NLFEM

Deflection/t

0.1

0

-0.1

-0.2

-0.3

-0.4
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

x/L

Figure 4.1 Analytical solution and FE solution

Differing adherend Young’s Modulus, E, thickness, t, length, L, bond length, 2c, and
adhesive thickness, η, were all investigated. All but the case of varying adhesive
thickness were conducted with a zero adhesive thickness, to study the effects of
dissimilar adherends on the adherends themselves. Both tensile loading and three-point
bending simulations were performed, and diagrams of each setup can be seen in Figures
4.2 and 4.3.

22

Figure 4.2 Finite element boundary conditions in tension

Figure 4.3 Finite element boundary conditions in three point bending

Figure 4.2 outlines the dimensional symbols of the model, with L being adherend length
beyond the overlap, c being half of the overlap length keeping with Goland and
Reissner’s [1] convention, and t being thickness of the adherend. Points A and B were
found to be the points of critical stress concentration, and subsequently all instances of
the parametric study had stresses measured at these points and compared to the stresses
measured in the “basic case,” a homogeneous single-lap joint with zero adhesive
thickness.
23

Regarding the basic cases, the input stress in the tension case, σ0, was unity, in order to
make all results normalized with respect to this input stress. The input load, for the threepoint bending basic case, was unity as well, for the same reason. In both basic cases, L
was 15 m, c was 1.5 m, and t was 1 m. Young’s Modulus and Poisson’s Ratio in each
case was set to near that of aluminum, 70 GPa and 0.33 respectively. The mesh size used
was the same uniform, isoparametric mesh used in the verification case, in order to hold
the basic cases and parametric study cases to follow to that verification. As is illustrated
in Figure 4.2, the far positive x face of the lower adherend was pinned at its midpoint,
and the far negative x face, where the input stress was applied, was assigned a roller
condition at its midpoint, again catering to the conditions in [1]. Contour maps of the
critical region and tables of stress values at points A and B for these basic cases are
shown below. Countour maps are rotated 90 degrees clockwise in order to maximize the
size of the map in the allowable space on the page. A rotated coordinate system is
transposed over the image for ease of understanding. The Von-Mises stress in ABAQUS
is calculated according to equation 4.1, while the other stresses are simply the classic
definition.

2
2
2
 vm   xx   xx yy   2  3 xy
yy

24

(4.1)

Table 4.1 Tensile basic case stress values at points A and B
A

B

σvm

4.996

7.317

σxx

2.379

7.368

σyy

3.927

1.962

τxy

-2.099 -1.813

25

Figure 4.4 Von-Mises stress contour map under unit tensile loading
(For the interpretation of the references to color in this and all other figures, the
reader is referred to the electronic version of this thesis)

26

Figure 4.5 X direction stress contour map under unit tensile loading

27

Figure 4.6 Y direction stress contour map under unit tensile loading

28

Figure 4.7 XY shear stress contour map under unit tensile loading

29

Table 4.2 Tensile basic case stress values at points A and B
A

B

σvm

31.15

98.10

σxx

16.25 -98.04

σyy

27.38 -55.74

τxy

-11.66

30

25.64

Figure 4.8 Von-Mises stress contour map under three point bending load

31

Figure 4.9 X direction stress contour map under three point bending load

32

Figure 4.10 Y direction stress contour map under three point bending load

33

Figure 4.11 XY shear stress contour map under three point bending load

As can be immediately seen from the stress contours, there are two obvious concentration
points on each adherend in an ideally bonded single-lap joint. The following chapters
explore how these points’ stresses vary as various parameters are changed, and discuss
why.

34

REFERENCES

35

REFERENCES

[1] Tsai, M. Y., Morton, J. “An Evaluation of Analytical and Numerical Solutions to the
Single-Lap Joint.” Int. J. Solids Structures. 31. (1994)
[2] Hart–Smith, L. J. “Adhesive-bonded single-lap joints.” NASA. CR-112236. (1973)
[3] Oplinger, D. W. “A layered beam theory for single lap joints.” Army Materials
Technology Laboratory Report. MTL TR91-23. (1991)
[4] Goland, M., Reissner, E. “The stresses in cemented joints.” J. Appl. Mech. 11. (1944)
[5] Grant, L. D. R., Adams, R. D., de Silva, L. F. M. “Experimental and numerical
analysis of single-lap joints for the automotive industry.” International Journal of
Adhesion and Adhesive. 29. (2009)

36

5 Results of the FEA Based Parametric Study

5.1 Varied Adherend Thickness Under Ideal Bonding and a Tensile Load

Several key parameters of the single-lap joint, Young’s Modulus, adherend thickness,
bond length, free adherend length, and later adhesive thickness and Young’s Modulus,
were varied in order to ascertain the stresses’ sensitivity and general behavior under their
variation. The first parameter studied was Young’s Modulus, E, under tensile loading.
Tables 5.1 and 5.2 display the critical stresses and their stress concentration factor (SCF)
with respect to the basic case, outlined in the previous chapter, at points A and B
respectively, and Figures 5.2 and 5.3 show each SCF as the ratio of E to the basic case,
E0, varies. In each case, ratios of 0.2, 0.5, 2, and 5 were considered. Figure 5.1 highlights
the critical points and the varied parameters. Additional ratios were considered where
more data was needed in certain cases. Again, all stresses are normalized with respect to
their unit input load, and a uniform isoparametric mesh was used. This mesh has been
verified in the previous chapter with reference to the analytical displacement calculated.

Stresses were calculated at the centroid of each quadrilateral element at the critical points
A and B. Take note that only the lower adherend’s properties were varied, and all stress
measurements were taken from this lower adherend, unless otherwise noted. The ratios
studied rendered checking both adherends redundant, because as the ratio is increased,
the measured adherend played the roll of the fixed parameter adherend.

37

E0
A

B

E

Figure 5.1 Ideally bonded Young’s Modulus variation critical points

Table 5.1 Reference stress values and SCFs at point A with varying E/E0
E/E0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

2.444 (0.489)

0.964 (0.314)

2.294 (0.584)

-0.837 (0.399)

0.5

3.917 (0.784)

1.943 (0.634)

3.446 (0.878)

-1.457 (0.694)

1.0

4.996 (1.000)

3.066 (1.000)

3.927 (1.000)

-2.099 (1.000)

2.0

5.967 (1.194)

4.512 (1.472)

3.561 (0.907)

-2.697 (1.285)

5.0

8.358 (1.673)

6.780 (2.211)

2.264 (0.577)

-3.372 (1.607)

Table 5.2 Reference stress values and SCFs at point B with varying E/E0
E/E0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

9.290 (1.270)

9.214 (1.251)

2.817 (0.747)

-2.755 (1.520)

0.5

8.339 (1.140)

8.306 (1.127)

3.765 (0.999)

-2.189 (1.207)

1.0

7.317 (1.000)

7.368 (1.000)

3.769 (1.000)

-1.813 (1.000)

2.0

6.217 (0.850)

6.355 (0.863)

3.211 (0.852)

-1.340 (0.739)

5.0

5.028 (0.687)

5.206 (0.707)

2.119 (0.562)

-0.776 (0.428)

38

SCF at point A as E/E0 varies
2.5



2



vm
xx
yy



xy

SCF

1.5

1

0.5

0

0

1

2

3

4

E/E0

Figure 5.2 SCFs as E/E0 varies at point A

39

5

SCF at point B as E/E0 varies
2


1.8




1.6

vm
xx
yy



xy

SCF

1.4
1.2
1
0.8
0.6
0.4

0

1

2

3

4

5

E/E0

Figure 5.3 SCFs as E/E0 varies at point B

Observing the SCFs at point A, for each type of stress except the Y direction, they
increase as the ratio E/E0 increases. At point B, however, the opposite is true. As the ratio
increases, the SCFs decrease, although not as drastically. For both points, Y direction
SCF increased up to a ratio of unity, and then decreased. X direction stress is the most
sensitive to variations in adherend E at point A, while XY shear stress was the most
sensitive at point B. Therefore, one should maintain an E/E0 ratio of unity across both
adherends when constructing a single-lap joint for tensile loading.

40

Before commenting on why the various stresses behave in the manner that they do, one
must comment on the definition of SCF in these cases. SCF is simply the ratio of current
stress at the point in question to the stress measured in the basic, homogeneous case.
Therefore, increasing SCF can be attributed to an increasingly tensile stress if basic case
stress is positive, or increasingly compressive stress if basic case stress is negative. One
must observe that Von Mises stress is always positive, and therefore is simply a way to
visualize stress evolution across X, Y, and XY directions in a single value. In reference to
varied Young’s Modulus, for Von Mises stress and SCF is always positive as expected. X
and Y direction stresses and SCFs are also always positive. XY direction shear stress is
always negative, but SCF is always positive, as expected. Therefore, for X and Y
direction stresses, increasing SCF means an increasingly tensile stress, and decreasing
refers to decreasingly tensile stress, or increasingly compressive stress. In regards to
decreasing SCF, the stress is simply becoming less tensile if it approaches zero. If SCF
crosses zero, the stresses are shifting from tensile to compressive.

As single-lap joints are loaded according to the boundary conditions displayed in the
previous chapter, in tension, the bonded region tends to rotate counter-clockwise due to
the eccentric loading angle. Therefore, Y direction stresses at point A become
increasingly tensile with increased transferred load, and at point B become increasingly
compressive with increased load. The same holds true for X direction stress, although this
is also influenced by the normal load applied to the upper adherend. Shear stress, due to
sign convention, increases in negative intensity at point A, and decreases in negative
intensity at point B, due to the direction of the load.

41

In reference to varied Young’s Modulus, all stresses are explained by this logic except for
Y direction stress. At both points, it increases up to a Young’s Modulus ratio of unity, but
then decreases as the ratio increases away from unity. At point A, this means it follows
the previous logic up to a ratio of unity, and then diverges. At point B, this means it
diverges from the previous logic up to a ratio of unity, and then converges. For ratios less
than unity, the varied adherend, from which the stresses are measured, has a smaller
stiffness than the unvaried adherend. This means that it will displace more, and
experience less stress. For ratios greater than unity, the opposite is true. However,
whether the ratio is less than or greater than unity, there is always one adherend that has a
higher Young’s Modulus than the other, and therefore it has greater stiffness. This
stiffness resists the rotation of the bonded region, and therefore, the moment caused by it,
resulting in a lesser Y direction stress regardless of Young’s Modulus ratio, and the
stresses were instead translated into X direction and XY shear stresses. In other words, Y
direction stress is decreased because regardless of the Young’s Modulus ratio, except if
unity, one adherend is always stiffer than the other. This stiffer adherend dictates the
amount of central rotation, and the rotation is translated into axial deformation of the less
stiff adherend. A ratio of unity is different, as the body is ideally bonded, and acts as a
homogeneous structure.

42

5.2 Varied Adherend Thickness Under Ideal Bonding and a Tensile Load

Varying adherend thickness was also investigated. Tables 5.3 and 5.4 show reference
stress values and SCFs at points A and B under tensile loading, and Figures 5.5 and 5.6
plot each stress’ evolution as the ratio of t/t0 is varied. Figure 5.4 highlights the critical
points and parameter being varied in this case.

t0
t
A

B

Figure 5.4 Ideally bonded thickness variation critical points

Table 5.3 Reference stress values and SCFs at point A with varying t/t0
t/t0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

2.692 (0.539)

1.918 (0.626)

1.645 (0.419)

-1.207 (0.575)

0.5

3.738 (0.748)

2.418 (0.789)

2.780 (0.708)

-1.629 (0.776)

1.0

4.996 (1.000)

3.066 (1.000)

3.927 (1.000)

-2.099 (1.000)

2.0

7.200 (1.441)

4.281 (1.396)

5.825 (1.483)

-2.955 (1.408)

5.0

13.890 (2.780)

7.930 (2.586)

11.610 (2.957)

-5.527 (2.633)

43

Table 5.4 Reference stress values and SCFs at point B with varying t/t0
σvm (SCF)

t/t0

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

50.320 (6.877) 52.400 (7.112)

19.79 (5.251)

-9.168 (5.057)

0.5

15.560 (2.127) 15.860 (2.153)

7.285 (1.933)

-3.528 (1.946)

1.0

7.317 (1.000)

7.368 (1.000)

3.769 (1.000)

-1.813 (1.000)

2.0

3.752 (0.513)

3.756 (0.510)

1.991 (0.528)

-0.961 (0.530)

5.0

1.217 (0.166)

1.245 (0.169)

0.420 (0.111)

-0.252 (0.139)

SCF at point A as t/t0 varies
3
2.5

SCF

2
1.5


1




0.5

vm
xx
yy



xy

0

0

1

2

3
t/t0

Figure 5.5 SCFs as t/t0 varies at point A

44

4

5

SCF at point B as t/t0 varies
8


7




6

xx
yy



xy

5

SCF

vm

4
3
2
1
0

0

1

2

3

4

5

t/t0
Figure 5.6 SCFs as t/t0 varies at point B
Varied adherend thickness behaved much the same as varied Young’s Modulus. An
increasing ratio of t/t0 resulted in increasing SCFs in all stresses at point A, and
decreasing SCFs in all stresses at point B. All stresses at each point behaved roughly the
same, however stresses at point B were impacted much more by smaller adherend
thickness. As with the variation in E, a t/t0 ratio of unity is suggested for construction
single-lap joints in tension.

The logic displayed after the case of varied Young’s Modulus still holds here, and
explains the universal increase in SCF at point A and decrease in SCF at point B.
Regarding Y direction stress, the varied thickness not only impacts adherend stiffness,
45

but effects the moment of inertia on a cubic level, and also increases or decreases the
eccentricity of the load, and therefore the moment. Whether the varied adherend is
thinner or thicker than the unvaried adherend, the eccentricity of the load is largely
affected. This moment causing eccentricity has been shown to largely impact the edge
moment in the bonded region, and as this is the dominant stress, it overshadows all other
effects thickness may have. As a result, Y direction stress follows the same trends as all
other forms of stress measured, and all stresses behave as expected under varied
thickness.

46

5.3 Varied Free Adherend Length Under Ideal Bonding and a Tensile Load

Free adherend length variation was then simulated. Tables 5.5 and 5.6 tabulate reference
stress values and their associated SCFs as L/L0 varies, and Figure 5.8 and 5.9 illustrate the
SCFs graphically. Figure 5.7 highlights the parameter under variation and the critical
points investigated.

L0
L
A

B

Figure 5.7 Ideally bonded free adherend length variation critical points

Table 5.5 Reference stress values and SCFs at point A with varying L/L0
L/L0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

8.530 (1.707)

5.062 (1.651)

6.907 (1.759)

-3.499 (1.667)

0.5

6.973 (1.396)

4.183 (1.364)

5.595 (1.425)

-2.883 (1.374)

1.0

4.996 (1.000)

3.066 (1.000)

3.927 (1.000)

-2.099 (1.000)

2.0

3.868 (0.774)

2.428 (0.792)

2.974 (0.757)

-1.651 (0.787)

5.0

2.327 (0.466)

1.550 (0.506)

1.663 (0.424)

-1.036 (0.494)

47

Table 5.6 Reference stress values and SCFs at point B with varying L/L0
L/L0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

0.692 (0.095)

0.550 (0.075)

-0.291 (-0.077)

-0.050 (0.028)

0.5

4.242 (0.580)

4.287 (0.582)

2.027 (0.538)

-1.016 (0.560)

1.0

7.317 (1.000)

7.368 (1.000)

3.769 (1.000)

-1.813 (1.000)

2.0

9.074 (1.240)

9.128 (1.240)

4.765 (1.264)

-2.268 (1.251)

5.0

11.491 (1.571)

11.548 (1.567)

6.134 (1.628)

-2.894 (1.596)

SCF at point A as L/L0 varies
2


1.8




1.6

vm
xx
yy



xy

SCF

1.4
1.2
1
0.8
0.6
0.4

0

1

2

3

4

L/L0

Figure 5.8 SCFs as L/L0 varies at point A

48

5

SCF at point B as L/L0 varies
2

1.5

SCF

1

0.5





0

vm
xx
yy



xy

-0.5

0

1

2

3

4

5

L/L0
Figure 5.9 SCFs as L/L0 varies at point B

Under varied free adherend length, SCFs at points A and B behave with trends opposite
that of varied Young’s Modulus or thickness. At point A, the SCFs decrease with
increasing L/L0, and at point B, the SCFs increase. Although the stress’ behavior is
opposite the varied E and t cases, one should still attempt to keep the two adherends in a
single-lap joint of equal length in order to minimize overall adherend stress in tension.

Free adherend length affects the eccentricity of the load, and therefore the edge moment,
which dominates the bonded region’s stresses. When the ratio is less than unity, it greatly
effects the eccentricity, but larger than unity, it effects it less and less. This is reflected, at
49

both points A and B, by the large slopes for ratios less than unity, and smaller slopes at
the ratio increases past unity. As free adherend length is shortened, point A’s stresses
become more tensile, and point B’s stresses become less tensile. As expected, the
opposite trend is observed as free adherend length is increased.

50

5.4 Varied Bonding Length Under Ideal Bonding and a Tensile Load

Joint bonding length was also varied. Tables 5.7 and 5.8 display the reference stress
values and SCFs at points A and B respectively, and Figures 5.11 and 5.12 display the
evolving SCFs graphically. Figure 5.10 highlights the varied parameter and critical
points.

2c

A

B

Figure 5.10 Ideally bonded bonding length variation critical points

Table 5.7 Reference stress values and SCFs at point A with varying c/c0
c/c0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.20

5.100 (1.021)

3.337 (1.088)

2.849 (0.726)

-2.347 (1.118)

0.50

4.789 (0.959)

3.083 (1.006)

3.558 (0.906)

-2.089 (0.995)

0.75

5.006 (1.002)

3.087 (1.007)

3.914 (0.997)

-2.112 (1.006)

1.00

4.996 (1.000)

3.066 (1.000)

3.927 (1.000)

-2.099 (1.000)

1.50

4.641 (0.923)

2.856 (0.932)

3.637 (0.926)

-1.954 (0.931)

2.00

4.599 (0.921)

2.841 (0.927)

3.592 (0.915)

-1.942 (0.925)

5.00

3.861 (0.773)

2.425 (0.791)

2.965 (0.755)

-1.649 (0.786)

51

Table 5.8 Reference stress values and SCFs at point B with varying c/c0
c/c0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.20

8.120 (1.110)

8.251 (1.120)

3.064 (0.813)

-1.808 (0.997)

0.50

7.424 (1.015)

7.514 (1.020)

3.491 (0.926)

-1.760 (0.971)

0.75

7.372 (1.008)

7.427 (1.008)

3.764 (0.999)

-1.820 (1.004)

1.00

7.317 (1.000)

7.368 (1.000)

3.769 (1.000)

-1.813 (1.000)

1.50

6.813 (0.931)

6.861 (0.931)

3.497 (0.928)

-1.687 (0.931)

2.00

6.785 (0.927)

6.837 (0.928)

3.458 (0.918)

-1.672 (0.922)

5.00

5.805 (0.793)

5.860 (0.795)

2.875 (0.763)

-1.409 (0.777)

SCF at point A as c/c 0 varies
1.3



1.2



1.1

vm
xx
yy



xy

SCF

1
0.9
0.8
0.7

0

1

2

3

4

c/c0

Figure 5.11 SCFs as c/c0 varies at point A

52

5

SCF at point B as c/c 0 varies
1.15


1.1




1.05

xx
yy



xy

1

SCF

vm

0.95
0.9
0.85
0.8
0.75

0

1

2

3

4

5

c/c0

Figure 5.12 SCFs as c/c0 varies at point B
The varied bond length cases behave differently than the previously studied varied
parameters. Prior to a ratio of 1.5, they seem erratic. The stresses with the most dramatic
variation are X direction stress at point A and Y direction stress at point B. Following
that, however, they seem to universally decrease with increased bond length. Therefore,
one should try to maximize bond length in order to minimize overall stress in the singlelap joint.

Bonding length changes effect the SCFs differently than the other varied parameters. For
ratios much less than unity, for the geometry used in these simulations, points A and B
begin to affect each other. This is shown in Figures 5.7 and 5.8 by the seemingly strange
53

behavior at ratios less than unity. However, as ratios increase past unity, the two points’
stress states become more independent of each other, and stresses universally decrease at
each point. This can be attributed to the simple fact that the load is dispersed across a
larger amount of material, with a larger cross-sectional area.

54

5.5 Varied Young’s Modulus Under Ideal Bonding and a Three-Point Bending Load

The same important lap joint parameters were also varied under a three-point bending
load. Tables 5.9 and 5.10 display the reference stresses and SCFs at points A and B, while
Figures 5.14 and 5.15 graphically present the change in SCF with respect to the ratio of
E/E0. Figure 5.13 highlights the varied parameter and critical points.

E0
A

B

E

Figure 5.13 Ideally bonded Young’s Modulus variation critical points

Table 5.9 Reference stress values and SCFs at point A with varying E/E0
E/E0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

17.19 (0.552)

6.057 (0.373)

16.55 (0.605)

-5.352 (0.459)

0.3

20.60 (0.661)

7.95 (0.571)

19.55 (0.714)

-6.579 (0.637)

0.5

25.17 (0.808)

10.99 (0.676)

23.29 (0.851)

-8.486 (0.728)

0.75

28.75 (0.923)

13.92 (0.857)

25.89 (0.946)

-10.33 (0.886)

1.0

31.15 (1.000)

16.25 (1.000)

27.38 (1.000)

-11.66 (1.000)

2.0

36.27 (1.164)

22.56 (1.388)

29.50 (1.077)

-14.68 (1.259)

5.0

42.03 (1.349)

31.29 (1.926)

30.14 (1.101)

-17.77 (1.524)

55

Table 5.10 Reference stress values and SCFs at point B with varying E/E0
E/E0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

126.8 (1.293)

-123.9 (1.263)

-53.98 (0.968)

37.20 (1.451)

0.3

121.35 (1.237)

-118.9 (1.213)

-57.20 (1.026)

34.52 (1.346)

0.5

112.6 (1.148)

-111.0 (1.132)

-59.01 (1.059)

31.49 (1.148)

0.75

104.4 (1.064)

-103.7 (1.058)

-57.94 (1.040)

28.26 (1.102)

1.0

98.10 (1.000)

-98.04 (1.000)

-55.74 (1.000)

25.64 (1.000)

2.0

82.80 (0.844)

-84.24 (0.859)

-46.44 (0.833)

18.86 (0.736)

5.0

66.49 (0.678)

-68.75 (0.701)

-30.79 (0.552)

11.07 (0.432)

SCF at point A as E/E0 varies
2
1.8
1.6

SCF

1.4
1.2
1


0.8

vm

xx

0.6



0.4

yy



xy

0.2

0

1

2

3

4

E/E0

Figure 5.14 SCFs as E/E0 varies at point A

56

5

SCF at point B as E/E0 varies
2


1.8



xx

yy

1.6



xy

1.4
SCF

vm

1.2
1
0.8
0.6
0.4

0

1

2

3

4

5

E/E0

Figure 5.15 SCFs as E/E0 varies at point B

As can easily be seen, the results from the three-point bending case are very similar in
behavior to the tensile case. At point A, the SCFs universally increase with an increased
ratio of moduli, while at point B, they decrease overall. The same conclusion can be
drawn as well: one should hold Young’s Moduli homogeneous across adherends in order
to minimize overall stress.

Under a three-point bending load, the edge moment is much more pronounced. The same
Y direction stress phenomena observed under tensile variation of Young’s Modulus can
be seen in this case at point B. And at point A, rather than Y direction stress decreasing
57

past unity, it simply levels off. The same logic used to explain stresses under a tensile
load can be used here as well, because both three-point bending and tensile loading
induce an edge moment on the bond, which has been shown to be the dominant load
contributing to stress in single-lap joints.

58

5.6 Varied Adherend Thickness Under Ideal Bonding and a Three-Point-Bending
Load

Tables 5.11 and 5.12 display the reference stress values and associated SCFs for varied
t/t0 and figures 5.17 and 5.18 graphically display the evolving SCFs with respect to
varied adherend thickness ratio. Figure 5.16 highlights the varied parameter and critical
points.

t0
t
A

B

Figure 5.16 Ideally bonded thickness variation critical points

Table 5.11 Reference stress values and SCFs at point A with varying t/t0
t/t0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

21.14 (0.679)

-15.25(-0.939)

-11.61(-0.424)

9.331(-0.800)

0.5

22.81 (0.732)

12.53 (0.771)

19.30 (0.705)

-8.956 (0.768)

1.0

31.15 (1.000)

16.25 (1.000)

27.38 (1.000)

-11.66 (1.000)

2.0

24.79 (0.796)

12.07 (0.743)

22.59 (0.825)

-8.752 (0.751)

5.0

10.78 (0.346)

4.764 (0.293)

10.17 (0.371)

-3.527 (0.303)

59

Table 5.12 Reference stress values and SCFs at point B with varying t/t0
t/t0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

1290 (13.150)

-1343 (13.70)

-539.0 (9.670)

242.1 (9.442)

0.5

294.0 (2.997)

-298.0 (3.040)

-154.1 (2.765)

70.65 (2.756)

1.0

98.10 (1.000)

-98.04 (1.000)

-55.74 (1.000)

25.64 (1.000)

2.0

33.39 (0.340)

-33.07 (0.337)

-19.83 (0.356)

9.130 (0.356)

5.0

10.62 (0.123)

-10.32 (0.105)

-7.055 (0.127)

3.151 (0.123)

SCF at point A as t/t0 varies
1

SCF

0.5

0



-0.5



vm
xx
yy



xy

-1

0

1

2

3

4

t/t0
Figure 5.17 SCFs as t/t0 varies at point A

60

5

SCF at point B as t/t0 varies
14



12



10

vm
xx
yy



xy

SCF

8
6
4
2
0
0

1

2

3

4

5

t/t0

Figure 5.18 SCFs as t/t0 varies at point B
For the three-point bending load cases, varied thickness resulted in different SCF
behavior than it did in tension. At point A, the SCFs increased up to a ratio of unity, and
then decreased, while at point B, they universally decreased. However, this does not
mean that one should maximize thickness. The SCFs of a ratio of less than unity, at least
at point B, still suggest one should keep a ratio of unity, as any nonhomogeneity would
increase the stress in the adherend with the smallest thickness. Also, the changing
thickness greatly changes the stiffness and moment of inertia of the varied adherend,
resulting in a much different state of stress at the lower ratios.

61

The apparent odd behavior of stresses at point A can be explained by the fact that stresses
simply changed direction as compared to the other cases. This is indicated by the
negative SCFs. Von Mises stress, being always positive, shows the general stress
behavior regardless of sign, and they fit the general trend at point A. Stress there
increases up to a ratio of unity, and then decrease. However, it is apparent from the SCFs
that stresses at point B are much more critical, with SCFs reaching over 13. They
decrease as thickness increases because of the increased cross-sectional area and moment
of inertia. The behavior of stresses at point A for ratios less than unity is quite apparently
attributable to a shift from tensile to more compressive stresses. The compressive stresses
are still of appreciable magnitude at point A, and can be considered to increase in
compressive strength, just as point B increases in tensile strength, for ratios less than
unity.

62

5.7 Varied Free Adherend Length Under Ideal Bonding and a Three-Point-Bending
Load

Varied length was also investigated under three point bending. Tables 5.13 and 5.14
display reference stress values and their associated SCFs, while Figures 5.20 and 5.21
graphically track the changing SCFs with respect to the ratio L/L0. Figure 5.19 highlights
the varied parameter and critical points.

L0
L
A

B

Figure 5.19 Ideally bonded free adherend length variation critical points

Table 5.13 Reference stress values and SCFs at point A with varying L/L0
L/L0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

4.171 (0.134)

2.127 (0.131)

3.705 (0.135)

-1.537 (0.132)

0.5

13.12 (0.421)

6.691 (0.412)

11.66 (0.426)

-4.830 (0.414)

1.0

31.15 (1.000)

16.25 (1.000)

27.38 (1.000)

-11.66 (1.000)

2.0

49.88 (1.601)

26.65 (1.640)

43.30 (1.581)

-18.98 (1.628)

5.0

99.47 (3.193)

55.34 (3.406)

84.35 (3.081)

-38.96 (3.341)

63

Table 5.14 Reference stress values and SCFs at point B with varying L/L0
L/L0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

0.069 (7E-4)

-0.024 (2E-4)

0.056 (-0.001)

2E-4 (1E-4)

0.5

38.42 (0.392)

-38.16 (0.389)

-22.80 (0.409)

10.37 (0.404)

1.0

98.10 (1.000)

-98.04 (1.000)

-55.74 (1.000)

25.64 (1.000)

2.0

136.0 (1.386)

-136.0 (1.387)

-76.96 (1.381)

35.39 (1.381)

5.0

174.1 (1.775)

-174.0 (1.775)

-99.59 (1.787)

45.50 (1.775)

SCF at point A as L/L0 varies
3.5
3
2.5

SCF

2


1.5




1

vm
xx
yy



xy

0.5
0
0

1

2

3

4

L/L0
Figure 5.20 SCFs as L/L0 varies at point A

64

5

SCF at point B as L/L0 varies
2

1.5

SCF

1



0.5




0

vm
xx
yy



xy

-0.5

0

1

2

3

4

5

L/L0

Figure 5.21 SCFs as L/L0 varies at point B
As is immediately apparent, with increased L, all SCFs universally increase. This is an
intuitive response as three-point bending induces a large edge moment, and increased
length increases the moment arm, therefore increasing the stress.

This behavior is easy to explain. As free adherend length is increased, the moment arm is
increased, and therefore stresses at A and B are increased across the board.

65

5.8 Varied Bonding Length Under Ideal Bonding and a Three-Point-Bending Load

Bond length was then investigated. Tables 5.15 and 5.16 present reference stress values
and SCFs as bond length varies, and figure 5.23 and 5.24 graphically track varying SCFs
with varying bond length. Figure 5.22 highlights the varied parameter and critical points.

2c

A

B

Figure 5.22 Ideally bonded bonding length variation critical points

Table 5.15 Reference stress values and SCFs at point A with varying c/c0
c/c0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

151.9 (4.876)

45.27 (2.786)

153.7 (5.614)

-36.87 (3.162)

0.5

43.81 (1.406)

19.80 (1.219)

40.97 (1.496)

-14.64 (1.256)

1.0

31.15 (1.000)

16.25 (1.000)

27.38 (1.000)

-11.66 (1.000)

2.0

29.52 (0.947)

15.37 (0.946)

25.96 (0.948)

-11.03 (0.946)

5.0

31.29 (1.005)

16.18 (0.996)

27.62 (1.009)

-11.64 (0.998)

66

Table 5.16 Reference stress values and SCFs at point B with varying c/c0
c/c0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

182.8 (1.863)

-170.5 (1.739)

-161.1 (2.890)

61.55 (2.401)

0.5

111.9 (1.141)

-111.2 (1.134)

-67.95 (1.219)

30.29 (1.181)

1.0

98.10 (1.000)

-98.04 (1.000)

-55.74 (1.000)

25.64 (1.000)

2.0

89.81 (0.916)

-89.74 (0.915)

-51.13 (0.917)

23.49 (0.916)

5.0

91.81 (0.936)

-91.79 (0.936)

-51.91 (0.931)

23.92 (0.933)

SCF at point A as c/c 0 varies
6



5



xx
yy



4

SCF

vm

xy

3
2
1
0

0

1

2

3

4

c/c0
Figure 5.23 SCFs as c/c0 varies at point A

67

5

SCF at point B as c/c 0 varies
3



2.5



vm
xx
yy



xy

SCF

2

1.5

1

0.5

0

1

2

3

4

5

c/c0

Figure 5.24 SCFs as c/c0 varies at point B

As is immediately apparent an obvious and intuitive, all stresses universally decrease
with increasing bond length. Increasing bond length essentially increases the thickness of
the central of the single-lap joint, which is most effected by the bending moment
imparted by the three point bending load. With a higher thickness, and therefore a higher
moment of inertia, the stresses decrease.

Varied bond length stress behavior is also very easy to explain. The cross-sectional area
and moment of inertia of the bonded region is increased as bonding length is increased,
and therefore, stresses due to the dominant moment are decreased universally.
68

5.9 Varied Adhesive Thickness Under Adhesive Bonding and a Tensile Load

While ideal bonding cases are important to understand the behavior of the adherends
under common loading cases, it is important to understand how the adhesive used in
bonding real-world single-lap joints effect the adherends as well. Figure 5.25 displays a
diagram of the boundary conditions and additional parameters involved in the adhesively
bonded single-lap joint parametric study.

Figure 5.25 Finite element boundary conditions and parameters

First, adhesive thickness was studied using the original tensile basic case’s dimensions
and properties. Base adhesive thickness was 0.1 m. Ec and vc were 28 GPa and 0.4
respectively, which is considered a moderately stiff adhesive. Tables 5.17 and 5.18
display the reference stress values and their associated SCFs as adhesive thickness was
varied, and Figures 5.27 and 5.28, graphically illustrate the changing SCFs with respect
to η/η0 at points A1 and B2, the critical points on each adherend displayed in Figure 5.17.

69

Points A2 and B1 are shown in the following section to always be less critical than those at
points A1 and B2. Figure 5.26 highlights the varied parameter and critical points.
η
A1
B2

Figure 5.26 Adhesively bonded adhesive thickness variation critical points

Table 5.17 Stress values and SCFs as η/η0 varies at point A1
η/η0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

7.323 (0.942)

7.370 (0.943)

3.795 (0.932)

-1.819 (0.937)

0.5

7.491 (0.964)

7.538 (0.964)

3.899 (0.958)

-1.866 (0.961)

1.0

7.772 (1.000)

7.817 (1.000)

4.072 (1.000)

-1.942 (1.000)

2.0

8.332 (1.072)

8.374 (1.071)

4.417 (1.085)

-2.095 (1.079)

5.0

10.01 (1.288)

10.04 (1.284)

5.448 (1.338)

-2.552 (1.314)

Table 5.18 Stress values and SCFs as η/η0 varies at point B2
η/η0

σvm (SCF)

σxx (SCF)

σyy (SCF)

τxy (SCF)

0.2

7.323 (0.942)

7.370 (0.943)

3.795 (0.932)

-1.819 (0.937)

0.5

7.491 (0.964)

7.538 (0.964)

3.899 (0.958)

-1.866 (0.961)

1.0

7.772 (1.000)

7.817 (1.000)

4.072 (1.000)

-1.942 (1.000)

2.0

8.332 (1.072)

8.374 (1.071)

4.417 (1.085)

-2.095 (1.079)

5.0

10.01 (1.288)

10.04 (1.284)

5.448 (1.338)

-2.552 (1.314)

70

SCF at point A1 as /0 varies
1.4

1.3

SCF

1.2

1.1




1



vm
xx
yy



xy

0.9

0

1

2

3
 /

4

0

Figure 5.27 SCFs as η/η0 varies at point A1

71

5

SCF at point B2 as /0 varies
1.4

1.3

SCF

1.2

1.1




1



vm
xx
yy



xy

0.9

0

1

2

3
 /

4

5

0

Figure 5.28 SCFs as η/η0 varies at point B2
As adhesive thickness increases, the stresses at points A1 and B2 universally increase.
This is easily explainable, and is attributed to the increased eccentricity of the loading
angle, thereby increasing the edge moment force on the bonded region.

72

5.10 Varied Young’s Modulus Under Adhesive Bonding and a Tensile Load

In order to study adhesive effects on a single-lap joint with dimensions similar to that of
the experimental analysis in chapter six, the dimensions were changed to 6.35 mm by
155.575 mm adherends with a bonding length of 38.1 mm. The single-lap joints studied
prior to this were of a different ratio of length to thickness, and therefore behave slightly
differently. Young’s Modulus and Poisson’s Ratio are still 70 GPa and 0.3 respectively
for the adherends, and are still 28 GPa and 0.4 respectively for the adhesive. A unit input
stress was also applied, in order to keep the resulting stresses normalized with respect to
its input.

For the purpose of comparing this geometry with ideal bonding with that of adhesive
bonding, Table 5.21 displays the various stresses at the critical points on each adherend
without any adhesive and with ideal bonding, and Table 5.22 displays the various stresses
at each of the critical points with the aforementioned adhesive added, and their SCF with
respect to the values in Table 5.21.

Table 5.19 Stress values of the basic model without any adhesive
A1

B1

A2

B2

σvm

5.798

4.309

4.310

5.798

σxx

7.150

2.923

2.923

7.150

σyy

2.791

3.842

3.842

2.740

σxy

-2.037

-2.317

-2.317

-2.037

73

Table 5.20 Stress values of the basic model with adhesive, SCFs compared to table 1
A1

B1

A2

B2

σvm (SCF)

2.712 (0.367)

0.876 (0.203)

0.876 (0.203)

2.712 (0.367)

σxx (SCF)

3.376 (0.472)

0.561 (0.192)

0.561 (0.192)

3.376 (0.472)

σyy (SCF)

1.311 (0.470)

0.618 (0.161)

0.618 (0.161)

1.311 (0.470)

σxy (SCF)

-0.842 (0.413)

-0.473 (0.147)

-0.473 (0.147)

-0.842 (0.413)

As is immediately apparent, the addition of the adhesive increases the stresses across the
board, if it changes them at all. Most notably, shear stress is increased significantly at
points A1 and A2. However, the stresses at points A1 and B2 are always much higher than
at the other two points. Therefore, only these two critical points will be investigated.

Following this initial comparison, adherend Young’s Modulus was once again varied.
Tables 5.23, and 5.24 tabulate the stresses and associated SCFs with the variation, and
Figures 5.30, and 5.31 plot SCFs at each respective critical point as Young’s Modulus is
varied. Figure 5.29 highlights the varied parameter and critical points.

E0

A1
B2

E

Figure 5.29 Adhesively bonded Young’s Modulus variation critical points

74

Table 5.21 Stresses and SCFS at point A1 as E/E0 varies
E/E0
0.2
0.5
1
2
5

σvm (SCF)
3.919 (0.946)
4.044 (0.976)
4.143 (1.000)
4.225 (1.020)
4.303 (1.039)

σxx (SCF)
4.970 (0.948)
5.131 (0.979)
5.241 (1.000)
5.324 (1.016)
5.394 (1.029)

σyy (SCF)
1.317 (0.779)
1.595 (0.944)
1.689 (1.000)
1.675 (0.992)
1.543 (0.914)

σxy (SCF)
-0.882 (0.803)
-1.026 (0.934)
-1.098 (1.000)
-1.133 (1.032)
-1.140 (1.038)

Table 5.22 Stresses and SCFS at point B2 as E/E0 varies
E/E0
0.2
0.5
1
2
5

σvm (SCF)
5.049 (1.216)
4.560 (1.098)
4.153 (1.000)
3.812 (0.918)
3.515 (0.847)

σxx (SCF)
6.159 (1.171)
5.709 (1.086)
5.259 (1.000)
4.820 (0.917)
4.352 (0.827)

75

σyy (SCF)
1.738 (1.050)
1.757 (1.061)
1.656 (1.000)
1.376 (0.831)
0.926 (0.559)

σxy (SCF)
-1.667 (1.492)
-1.408 (1.261)
-1.117 (1.000)
-0.822 (0.735)
-0.500 (0.448)

SCF at point A1 as E/E0 varies
1.15
1.1
1.05

SCF

1
0.95
0.9

vm


0.85



0.8

xx
yy



xy

0.75

0

1

2

3

4

E/E0

Figure 5.30 SCF at point A1 as E/E0 varies

76

5

SCF at point B2 as E/E0 varies
2


1.8



xx

yy

1.6



xy

1.4
SCF

vm

1.2
1
0.8
0.6
0.4

0

1

2

3

4

5

E/E0

Figure 5.31 SCF at point B2 as E/E0 varies
Varying Young’s Modulus on this single-lap joint with moderate length adherends has
some interesting effects on the stresses at its critical points. Point A1 follows the same
trends in Y direction stress and all other stresses, and point B2 follows the Y direction
stress trend and all other stresses as the case with no adhesive, only with less magnitude.
This is due to the addition of the adhesive, and its ability to hold some of the stress that
was previously transferred directly from on adherend to the other.

77

5.11 Varied Adherend Thickness Under Adhesive Bonding and a Tensile Load

Tables 5.27, and 5.28 display the reference stress values and SCFs associated with varied
adherend thickness, and Figures 5.33, and 5.34 plot the SCF versus the varying thickness
ratio. Figure 5.32 highlights the varied parameter and critical points.
t0

t

A1
B2

Figure 5.32 Adhesively bonded adherend thickness variation critical points

Table 5.23 Stresses and SCFS at point A1 as t/t0 varies
t/t0
0.2
0.5
1
2
5

σvm (SCF)
3.565 (0.861)
3.523 (0.850)
4.143 (1.000)
5.553 (1.340)
9.838 (2.374)

σxx (SCF)
4.488 (0.856)
4.449 (0.849)
5.241 (1.000)
7.031 (1.342)
12.46 (2.379)

σyy (SCF)
1.029 (0.609)
1.289 (0.763)
1.689 (1.000)
2.450 (1.450)
4.728 (2.799)

σxy (SCF)
-0.771 (0.702)
-0.880 (0.802)
-1.098 (1.000)
-1.534 (1.398)
-2.847 (2.593)

Table 5.24 Stresses and SCFS at point B2 as t/t0 varies
t/t0
0.2
0.5
1
2
5

σvm (SCF)
19.93 (4.799)
8.956 (2.157)
4.153 (1.000)
2.121(0.511)
0.922 (0.222)

σxx (SCF)
23.68 (4.503)
11.47 (2.181)
5.259 (1.000)
2.661 (0.506)
1.149 (0.218)

78

σyy (SCF)
4.118 (2.487)
2.938 (1.773)
1.656 (1.000)
0.933 (0.564)
0.420 (0.253)

σxy (SCF)
-2.531 (2.266)
-2.022 (1.810)
-1.117 (1.000)
-0.632 (0.566)
-0.285 (0.256)

SCF at point A1 as t/t0 varies
3

2.5

SCF

2

1.5




1

vm
xx

yy


xy

0.5

0

1

2

3

4

t/t0
Figure 5.33 SCF at point A1 as t/t0 varies

79

5

SCF at point B2 as t/t0 varies
5



4

vm
xx

yy


xy

SCF

3

2

1

0

0

1

2

3

4

5

t/t0
Figure 5.34 SCF at point B2 as t/t0 varies

With varied adherend thickness, the SCFs at point A1 increase and at point B2 decrease as
t/t0 increases, exhibiting the same behavior as in the ideally bonded case. The same
explanation holds.

80

5.12 Varied Adhesive Young’s Modulus Under Adhesive Bonding and a Tensile
Load

Tables 5.26, and 5.27display the stresses and SCFs associated with varied adhesive
Young’s Modulus, and Figures 5.36, and 5.37 plot the changing SCF as adhesive
Young’s Modulus changes. Figure 5.35 highlights the varied parameter and critical
points.

Ec

A1
B2

Figure 5.35 Adhesively bonded adhesive Young’s Modulus variation critical points

Table 5.25 Stresses and SCFS at point A1 as Ec/Ec0 varies
Ec/Ec0
0.2
0.5
1
2
5

σvm (SCF)
3.632 (0.877)
3.883 (0.937)
4.143 (1.000)
4.467 (1.078)
4.966 (1.199)

σxx (SCF)
4.512 (0.861)
4.897 (0.934)
5.241 (1.000)
5.617 (1.072)
6.125 (1.169)

σyy (SCF)
1.104 (0.654)
1.463 (0.937)
1.689 (1.000)
1.811 (1.072)
1.831 (1.084)

σxy (SCF)
-0.600 (0.547)
-0.861 (0.785)
-1.098 (1.000)
-1.351 (1.231)
-1.664 (1.515)

Table 5.26 Stresses and SCFS at point B2 as Ec/Ec0 varies
Ec/Ec0
0.2
0.5
1
2
5

σvm (SCF)
3.633 (0.875)
3.888 (0.936)
4.153 (1.000)
4.480 (1.079)
4.980 (1.199)

σxx (SCF)
4.523 (0.860)
4.912 (0.934)
5.259 (1.000)
5.636 (1.071)
6.140 (1.167)
81

σyy (SCF)
1.086 (0.656)
1.437 (0.867)
1.656 (1.000)
1.774 (1.071)
1.852 (1.118)

σxy (SCF)
-0.612 (0.548)
-0.877 (0.785)
-1.117 (1.000)
-1.371 (1.228)
-1.679 (1.503)

SCF at point A1 as Ec/Ec0 varies
2
1.8
1.6

SCF

1.4
1.2
1

vm


0.8



0.6

xx
yy



xy

0.4

0

1

2

3

4

Ec/Ec0

Figure 5.36 SCF at point A1 as Ec/Ec0 varies

82

5

SCF at point B2 as Ec/Ec0 varies
2
1.8
1.6

SCF

1.4
1.2


1

vm

xx

0.8



0.6

yy



xy

0.4

0

1

2

3

4

5

Ec/Ec0

Figure 5.37 SCF at point B2 as Ec/Ec0 varies

As far as adherend stress is concerned, it is clear that the smallest adhesive Young’s
Modulus is best. Stresses at both critical points increase in intensity as the Young’s
Modulus of the adhesive increases, which is to be expected. A stiffer adhesive causes
more deformation and stress in the adherends.

83

5.13 Design Recommendations via Parametric Study
With a basic understanding of the effect each individual parameter has on the stresses in
the single-lap joint, one can develop recommendations concerning single-lap joint design.
Of course, the ideally bonded cases shed little light on “real world” lap joints, but the
corresponding cases simulated with an adhesive often held similar results, so design
recommendations based on the ideally bonded cases would be identical to those based on
the adhesively bonded cases.

1. As adhesive thickness increases, the stresses at the critical points A1 and B2
universally increase. Therefore, one should minimize the thickness of the
adhesive used in single-lap joint design.
2. Inspecting both varied Young’s Modulus and adherend thickness with an adhesive
bond, one sees that stresses at point A1 increase with an increase in each
parameter, and the stresses at point B2 decrease with an increase in each
parameter. Therefore, in order to keep stresses optimal, it is hypothesized that one
should keep the product of Young’s Modulus and adherend thickness, more
specifically the flexural rigidity, constant across adherends.
3. Concerning adhesive Young’s Modulus, as far as adherend stress is concerned,
one should minimize the adhesive Young’s Modulus, as the stresses at both
critical points increased with increasing adhesive Young’s Modulus. However, in
design, one must also consider the strength of the adhesive.

84

4. As bonding length increased, the stresses at both critical points decreased.
Therefore, it is recommended that one maximizes the bonding length of single-lap
joints within reason.
5. Concerning free adherend length, one should keep it constant across adherends.
The stress in the longer adherend was always found to be greater than that in the
shorter adherend.

85

6 Experimental Validation

Using an Instron 1321 tensile testing machine and load cells capable of measuring plus or
minus 100 kN load, two single-lap joint specimens were constructed of 6061 Aluminum
and FM-94 adhesive. One adherend was approximately twice the thickness of the other
adherend, as detailed in the measurements presented later. Due to the limited space in the
grips of the testing machine, some material was removed from the thicker adherend,
resulting in an exacerbated lateral deformation, also discussed later.

Digital Image Correlation (DIC) was used in order to track the full-field displacement of
the bonded region throughout tensile testing. A random black and white speckle pattern
was painted onto the bonding region which is used to track the movement of assigned
squares. The DIC software does this by assigning greyscale values to smaller regions that
the software discretizes the area of interest into. Each square contains a specified number
of pixels, and the greyscale values assigned to those pixels generates a unique pattern that
can be recognized and tracked by the Dantec software from image to image, allowing for
the generation of displacement data.

Figure 6.1 diagrams the approximate shape of the adherends, with letters assigned to each
side length and dimension in order to tabulate the dimensions of the two specimens in
Tables 6.1 and 6.2.

86

A

J

B
I
H

C

G

K

D
F

E

Figure 6.1 Diagram of the dimensions and shapes of the experimental specimens

87

Table 6.1 Dimensions of specimen 1
A

7.65 mm

B

35.00 mm

C 157.00 mm
D

6.46 mm

E

7.68 mm

F 151.66 mm
G

3.24 mm

H

11.00 mm

I

3.04 mm

J

26.00 mm

K

38.21 mm

88

Table 6.2 Dimensions of specimen 2
A

7.80 mm

B

28.60 mm

C 151.54 mm
D

6.37 mm

E

7.94 mm

F 147.80 mm
G

3.23 mm

H

2.77 mm

I

3.22 mm

J

25.83 mm

K

38.76 mm

Dimensions were measured to the best estimate possible, but some dimensions, like the
adhesive thickness, were so thin that incredible accuracy was impossible. For both
specimens, the best estimate possible was 0.003 mm. Several points were compared in Y
direction and X direction displacement for each specimen. These comparisons were made
at step 120 in DIC recording, with each step equating to 0.5 seconds. In the process of the
DIC software discretizing the area of interest, the outter edges of the area are cut off. In
order to account for this, points were chosen in reference to the adhesive, which is easy
enough to see in the DIC images. In the end, a decent correlation between DIC results
and finite element analysis results was possible.

89

Ideal bonding between adhesive and adherend was assumed, as well as 100% density of
both adherend and adhesive. Perfect square edges were also assumed, as well as a perfect
square termination of the adhesive. The same 40 element per thickness as verified in
previous sections was used, with biquadrilateral quadratic isoparametric plane strain
elements, with a measured thickness of 25.55 mm prescribed as the plane strain
thickness. The specimen was simulated vertically, as it was placed in the testing machine.
Upon inspection of the tested specimens, it was clear that the grips only gripped a portion
of the end tabs of the adherends. Therefore, an assumption of 80% tab length was made,
and on this length, the movement of the adherends was restricted to vertical movement
only. The top adherend’s top most corners were pinned. Finally, a central concentrated
load, measured by the testing machine’s load cell at step 120 was applied to the center of
the lower adherend’s bottom most face. A linear material model was used, but non-linear
geometry was taken into account.

Readily available material data states that the Young’s Modulus and Poisson’s Ratio of
6061 Aluminum are 68.9 GPa and 0.33, respectively. There is very limited data available
concerning the material properties of FM-94 adhesive, but [1] states that the Young’s
Modulus and Poisson’s Ratio of FM-94 are 1.9 GPa and 0.52, respectively. Both
materials were treated as elastic.

Curiously, preliminary results showed similar displacement in the Y direction, but were
off in the X direction. The only logical conclusion, was that the testing machine’s grips
were not perfectly centered. The DIC results made this apparent, by the location of a low

90

displacement semi-circle in the displacement magnitude contour plots. Therefore, a
lateral shift of 0.35 mm was applied to the bottom grip’s simulated boundary conditions.
When a lateral displacement was prescribed to the simulated lower grip, the results were
much closer to those of the DIC.

Because DIC outputs displacement, X and Y direction displacements were compared for
each specimen, at the points detailed in Figure 5.17, the thicker adherend’s points
denoted by subscript 1, and the thinner adherend’s points denoted by subscript 2. Table
6.3 displays the displacement values for specimen 1, and Table 6.4 displays the
displacement values for specimen 2. Figures 6.2, 6.3, and 6.4 display the specimen 1
finite element and DIC contours for displacement magnitude, X displacement, and Y
displacement respectively. Figures 6.5, 6.6, and 6.7 display the specimen 2 finite element
and DIC contours for displacement magnitude, X displacement, and Y displacement
respectively.

Table 6.3 Comparison of displacements for specimen 1
Ux,FEA (Ux,DIC) (mm) Uy,FEA (Uy,DIC) (mm) Ux Error Uy Error
A1

-0.181 (-0.196)

-0.255 (-0.320)

7.65%

20.31%

B1

0.811 (1.020)

-0.287 (-0.339)

20.49%

21.53%

A2

-0.182 (-0.193)

-0.266 (-0.335)

5.70%

20.60%

B2

0.811 (1.020)

-0.299 (-0.358)

20.49%

16.48%

91

Figure 6.2 Finite element (left) and DIC (right) displacement magnitude specimen 1

92

Figure 6.3 Finite element (left) and DIC (right) X direction displacement specimen 1

93

Figure 6.4 Finite element (left) and DIC (right) Y direction displacement specimen 1

94

Table 6.4 Comparison of displacements for specimen 2
Ux,FEA (Ux,DIC) (mm) Uy,FEA (Uy,DIC) (mm) Ux Error Uy Error
A1

0.155 (0.192)

-0.252 (-0.319)

19.27%

21.00%

B1

-0.842 (-0.964)

-0.286 (-0.466)

12.66%

38.63%

A2

0.156 (0.194)

-0.263 (-0.339)

19.59%

22.42%

B2

-0.842 (-0.964)

-0.299 (-0.481)

12.66%

37.84%

95

Figure 6.5 Finite element (left) and DIC (right) displacement magnitude specimen 2

96

Figure 6.6 Finite element (left) and DIC (right) X direction displacement specimen 2

97

Figure 6.7 Finite element (left) and DIC (right) Y direction displacement specimen 2

98

As can be seen, under the assumptions made for the finite element models, the
quantitative error between finite element and DIC results was about 20% for each
specimen, and qualitatively, the colormaps correlated fairly well. However, in specimen
2’s case, there were two points near 40% error. The same points, under similar load and
identical assumptions, in specimen 1, had an error of about 20%. Looking more at the
actual displacement values, the error is often less than 15 tenths of a millimeter. In the
case of the two largest percent errors, the difference is about 19 tenths of a millimeter.

These errors can be attributed to a variety of factors. The limited data on FM-94 adhesive
leaves the possibility of less than completely accurate material model. The assumption of
linear material behavior may be a contributing factor as well, but without time and more
samples, a nonlinear material model could not be obtained. In the case of the largest
percent errors, under similar loading, the finite element results were similar, but the DIC
results were significantly different. This leads one to believe the results obtained for Y
direction displacement obtained by DIC may have been affected other factors. For
example, the grips may have applied uneven pressure, the adhesive may have been less
than 100% dense, or the specimen may have slipped in the grip. A multitude of factors in
the experimental setup could have affected the DIC results, and testing more specimens
under higher levels of observation would shed light on this. A 100% density assumption
on the adhesive, along with the ideal bonding could also slightly effect the results.

DIC also grants some insight into the initiation of specimen failure. Applying the loads
associated with specimen failure, 22 kN for specimen 1 and 21.7 kN for specimen 2, the

99

values in Tables 6.4 and 6.5 were obtained at the critical points identified in the
parametric studies conducted on adhesively bonded single-lap joints in chapter 5.

Table 6.5 Failure stress predictions for specimen 1
σvm (MPa) σxx (MPa) σyy (MPa) τxy (MPa)
A1

599.143

319.029

711.610

-116.803

B2

758.960

369.960

916.931

-126.375

Table 6.6 Failure stress predictions for specimen 2
σvm (MPa) σxx (MPa) σyy (MPa) τxy (MPa)
A1

606.161

372.254

723.762

117.287

B2

792.588

326.845

939.834

130.468

As can be seen, under similar but slightly different geometry and loading, the stresses
measured are very similar at each point, with less than 10% difference in all stresses.
While reading the DIC images for failure initiation is not obvious or easy, a relatively
close estimate can be made, and the failure loads calculated for specimen are very
similar. If one were to obtain the failure stress of FM-94 adhesive, the material that fails
first during tensile testing with aluminum adherends, and find that this failure stress was
close to 900 MPa, one could verify that this simulation in fact accurately simulates the
specimen and can be used to predict failure in other simulated specimens.
100

REFERENCES

101

REFERENCES

[1] X. Zhang et. al. “Fail-safe design of integral metallic aircraft structures reinforced by
bonded crack retarders.” Engineering Fracture Mechanics. 76. (2009)

102

7 Conclusions
For each of the following conclusions, the reader is referred to the figures in chapter 5 for
the location of the critical points referenced.

1. Varied adherend Young’s Modulus, in both ideally bonded and adhesively bonded
single-lap joints with a tensile load applied, yielded increasing stress values as Young’s
Modulus increased at point A, and decreasing stress values as Young’s Modulus
increased at point B, except in reference to Y direction stress. Ratios anything different
than unity resulted in reduced Y direction stress as one adherend was always stiffer than
the other, and this caused the usual central bond rotation to be translated into axial
deformation of the less stiff adherend.

2. Varied adherend thickness, in both ideally bonded and adhesively bonded single-lap
joints with a tensile load, yielded varied stress values that increased with increasing
adherend thickness at point A, and stress values that decreased with increasing adherend
thickness at point B. While variations in adherend thickness varied stiffness in this case
as well as the case of varied Young’s Modulus, the eccentricity of the load and moment
of inertia of the varied adherend were also affected. This caused Y direction stresses to
follow the same trends as the other measured stresses, rather than following the
phenomena resulting from the varied Young’s Modulus case.

3. Free adherend length variation yielded different results than varied thickness or
Young’s Modulus. With increasing free adherend length, the stresses decreased in

103

intensity at point A and increased in intensity at point B. Its variation greatly affects the
eccentric loading angle for ratios less than unity, and affect it less and less as ratios
increase up past unity, which greatly affects stresses for the former, and less so for the
latter, a phenomena that was observable by increasing or decreasing slope in the SCF
plots.

4. Varied bonding length exhibited the expected behavior for ratios greater than unity; the
stresses at both critical points decreased as the ratio increased. For ratios much less than
unity, the stresses at each point began to affect the stresses at the other point, resulting in
seemingly strange results in the stresses. For ratios greater than unity, there was a
uniform decrease in all measured stresses as bonding length increased.

5. Varied adherend Young’s Modulus and bonding length under a three point bending
load exhibited the same behavior as under a tensile load. This is to be expected;
increasing Young’s Modulus yielded an increasing adherend stiffness, and increasing
bonding length increased the moment of inertia and general thickness of the member near
the application of the bending load, resulting in the same universal reduction in stress
intensity observed in the tensile loading case.

6. Varied adherend thickness under a three-point bending load produced much the same
results that varied bonding length produces; stresses decrease at both points under
increased thickness because there is a greater cross-sectional area and moment of inertia
as a result of this increased thickness. The variation in the right hand boundary

104

condition’s location could cause the increasing compressive intensity, rather than tensile
intensity, for ratios less than unity at point A.

7. Free adherend length variation under a three-point bending load behaved much as was
expected; increasing length increases the lateral moment arm of the applied force, and
therefore universally increases the stresses at both critical points.

8. The addition of an adhesive over the ideally bonded cases had one most notable effect;
it made the stresses at the point formerly designated B the most intense, and made those
at point A small to the point of unimportance.

9. As adhesive thickness increases, the stresses at both critical points increases. This is to
be expected, as it increases the eccentricity of load, and therefore the dominant force on
the central bonding region of the single-lap joint.

10. Increasing adhesive Young’s Modulus increases the stresses at both critical points, as
the adhesive increases in stiffness, deforms less, and forces more deformation and stress
on the adherends.

11. Using a linear material model and nonlinear geometry, along with approximate but
near accurate measurements and boundary condition mimicry and a verified mesh
density, one can predict the deformation of a single-lap joint in a tensile test with an

105

average of 20% error. This error can be reduced by a more accurate material model, more
accurate measurements, more accurate testing comparison, and a more ideal testing setup.

12. As one can simulate deformation of an experimental single-lap joint, they can also
predict the failure stress with a finite element model reasonable accurately. With accurate
indications as to the initiation of adhesive failure and more accurate testing, one can
improve the accuracy of this prediction.

106

8 Suggestions for Future Research

1. In order to more deeply understand the affect varied material and geometric
parameters have on the critical points’ stresses, one vary more than one parameter
at once, to understand how each parameter interacts with the other.

2. One could also more deeply understand single-lap joints if they were to study
exceedingly long and exceedingly short lap joints, rather than the joints of
intermediate length studied in this thesis.

3. In order to increase the accuracy of the experimental finite element model, one
can do several things:
a. Develop a more accurate strain-dependent material model for both the
adherend material and adhesive material.
b. Take steps to assure that the testing grips apply equal pressure to each
clamped region and that they are as perfectly aligned as possible.
c. Assure that the DIC setup is not disturbed during the experiment’s
duration, and assure that the camera used is as perfectly aligned as
possible.
d. Ensure that all dimensional measurements are as accurate as possible, and
that any machining that is needed is as accurate as possible, and that the
gripped regions can be perfectly aligned vertically.

107