18

TH
INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS
1 Introduction
With the increasing demand of energy and
environmental protection, composite materials have
been extensively used in aeronautic structures,
especially in the primary load components, due to its
weight-saving potential. Therefore, the section of
some composite structures could be much thicker
than before. Thick-section composites are ones
where the effect of geometry, material constituents,
lamination scheme, processing and service loading
exhibit three dimensional states of stress.
In order to join thick-section composites, it still
requires bonding, fastening or hybrid ways, among
which the bolted joint is a popular method because
of its high reliability, load-carrying capacity and
convenience to disassemble
[1]
. However, stress
concentration, composite brittleness and anisotropy
are sources of weakness in mechanically fastened
joints, what is more, the single-lap joint is more
dangerous because of the secondary bending effect
induced, and these disadvantages could be more
serious when thick-section laminates take part in the
joint. Thus, precise prediction of the load
distribution in bolted joint composite structures is in
bad need for the industry engineers and researchers.
Traditionally, there are 3 methods to study the load
distribution in multi-bolt composite joints:
experimental test methods, analytical method and
finite element analysis method. ASTM committee
has developed test procedure ASTM D7248 to
assess the load distribution of 2-fastener polymer
matrix composite laminates. The load distribution
can also be measured by instrumented bolt with
rosette strain gauges
[2]
. However, the test method is
expensive and time-consuming, so its not easy for
aircraft designer to adopt. Finite element analysis,
which usually requires a 3D nonlinear finite element
model since the load is not uniform through the
laminate thickness
[3,4]
, often costs dozens of hours to
yield the results.
This paper, considering the changes of fasteners
flexibility and plates flexibility introduced in thick
laminate single-lap joint, will introduce a new
analytical tool, which makes the designer in the
comfort of predicting the load distribution in multi-
bolt single-lap thick laminate joint.
2 Fastener Flexibility in Composite Joint
The flexibility of the mechanical composite joint is
composed of the flexibilities of the plates and
fasteners, and the co-operations of those flexibilities
will influence the load transfer over the fasteners. In
addition, the flexibilities of the plates are smaller in
thick laminate joint comparing with that in thin
laminate joint, which means the fastener flexibility
will take more effort in load sharing than that in thin
plates joint. Futhermore, the load carried by fastener
will not be uniform along the plate thickness in
single-lap joint, which will influence the fasteners
displacement, that is to say the flexibility of fastener
needed to be verified according to the load condition.
To be precise, the flexibilities of the fasteners in the
thick laminate single-lap joint needs to be more
accurately estimated according to the actual.
2.1 Load Condition of Single-lap Bolted Joint
The load condition of single-lap bolted composite
joint is shown in Fig.1. it has shown that the
fasteners undergo the bearing force (the fastener
exert bearing force to the plate inversely)from plates,
shearing force and bending moment introduced by
eccentric load from the two plates, thus the
deformation in the joint includes 6 parts: the
shearing deformation of the fastener, the bending
deformation of the fastener, the fasteners bearing
deformation by plate 1, the fasteners bearing
deformation by plate 2, the plate 1 bearing
AN ANALYTICAL TOOL TO PREDICT LOAD DISTRIBUTION
OF MULTI-BOLT SINGLE-LAP THICK LAMINATE JOINTS

Longquan Liu*, Ying Mao, Ran Wei
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China
* Corresponding author(liulongquan76@sjtu.edu.cn)

Keywords: thick laminate, analytical tool, load distribution, single-lap
deformation by the fastener, the plate 2 bearing
deformation by the fastener.

Fig.1. Load condition in single-lap joint
2.2 Fastener Flexibility over Uniform Bearing Load
1) Force analysis
The Fastener load model over uniform bearing load
is shown in Fig.3 (a), which is a statically
indeterminate system that can be divided into Fig.3
(b) and Fig.3(c).

(a)

(b) (c)
Fig.3. Fastener over uniform bearing load
The angle of rotation of the beam
1
under
1 1
t w ,
2
under
2 2
t w and
3
under the moment M
[5]
,
EI
t w
6
3
1 1
1
= (1)
EI
t L w
6
) (
3
1
3
2
2

= (2)
EI
ML
=
3
(3)
While the rotation angle of the beam on the right
equals zero,
0
3 2 1
= + = (4)
Equilibrium equation,
2 2 1 1
t w t w P = = (5)
Substituting
1
,
2
,
3
from Equation (1-3) to
Equation (4),
0
6
) (
6
3
1
3
2
3
1 1
= +

EI
ML
EI
t L w
EI
t w
(6)
Thus,
L
t t t t t w
M
6
) 3 2 (
3
2
2
2 1 2
2
1 2
+ +
= (7)
2) Flexibility
As shown in Fig.3, apart from undertaking
deformation caused by shear stress and bending
moment, the combination of fastener and plate also
endure the bearing deformation between the two
objects. Hence, the deformation between Point a and
Point b of the bolt is the result of the co-operation of
the six different constituents mentioned above. The
displacements of Point a caused by
1 1
t w
2 2
t w M,
(

+ =
2 1 1
1
2
1
2 1
1
1
)
2
(
2
4 6
24
)
2
(
t t
t t
EI
t
w
a

(8)
(

+ =
2
2 3 3
12
)
2
(
1
2 2 1 2
2 1
2
2
t
t t t L t
EI
t
w
a

(9)
EI
t
M
a
2
)
2
(
2 1
3
=
(10)
3 2 1 a a a a
+ + =
(11)
Displacements of Point b caused by
1 1
t w
2 2
t w M,
(

+ =
1
2
1
3
1 1
1
)
2
( 4
24
t
t
t
EI
t w
b
(12)
] )
2
( 4 )
2
( ) ( 6
)
2
)( ( 4 )
2
[(
24
4
1
2
1
3
1
2 2
1
2
2 1
3 2
1 2 1
4 2
1
2
2
t
t
t t
t
t t t
t
t t t
t
t
EI
w
b
+ + + + +
+ + + =
(13)
Plate 1
Plate 2

3
PAPER TITLE
EI
t
t M
b
2
)
2
(
2 2
1
3
+
= (14)
3 2 1 b b b b
+ + = (15)
The bending flexibility of the bolt,
LEI
t t t t t t t t
P P
C
a b
bb
384
9 57 96 57 9
4
2
3
2 1
2
2
2
1 2
3
1
4
1
+ + + +

= =

(16)
The shear flexibility, bearing flexibility of the bolt
and the plate flexibility,
b b
bs
A G
t t
C
9
) ( 4
2 1
+
= (17)
bbr bbr
bbr
E t E t
C
2 1
1 1
+ = (18)
2 2 1 1
1 1
x x
pbr
E t E t
C + = (19)
Thus the fasteners flexibility over uniform bearing
loads through substitution and calculation,
2 2 1 1 2 1
2 1
4
2
3
2 1
2
2
2
1 2
3
1
4
1 2 1
1 1 1 1
) ( 384
9 57 96 57 9
9
) ( 4
x x bbr bbr
b b
E t E t E t E t
t t EI
t t t t t t t t
A G
t t
F
+ + + +
+
+ + + +
+
+
=
(20)
2.3 Fastener Flexibility in Single-lap Bolted Joint
The contact force is non-uniform along the plate
thickness in the single-lap joints shown in Fig.3,
therefore the items in Equation (1) needed to be
corrected accordingly. Though they can be obtained
through fitting the experimental results, there is no
physical meaning on every factor, besides the factors
may change with different experimental results, and
the experimental cost is high. Several single-joint
3D finite element models, whose modeling method
is validate by the test results carried out according to
ASTM D 5961 test standard[6], employed the 6
correction factors separately.
The 3D finite element model of single lap bolted
composite joint and contact area setting are shown in
Fig.4.

(a)

(b)
Fig.4. 3D FE model of single-lap bolted joint and
contact bodies
Comparing the bearing deformation of the plates,
and the bearing deformation, the shear deformation
and bending deformation of the bolt separately,
correction factors of the single-lap thick laminate
bolted joint could be constructed respectively. The
fasteners flexibility in single-lap bolted thick
composite joint (named as Liu-Mao in this paper) is
shown as:
2 2 1 1 2 1
2 1
4
2
3
2 1
2
2
2
1 2
3
1
4
1 2 1
15 . 1 15 . 1 15 . 1 15 . 1
) ( 00 384
9 57 96 57 9
6
x x bbr bbr
b b
E t E t E t E t
t t EI
t t t t t t t t
A G
t t
F
+ + + +
+
+ + + +
+
+
=
(21)
3 Analytical Tool of Load Distribution in Multi-
bolt Joint
In aircraft structure design, except for seal areas
where the fasteners are staggered, joint between two
structures can be divided into several multi-row
single-column substructures. Usually, 3 rows is
utmost. Following contents focus on constructing the
2 row and 3 row fasteners joint analytical model.
3.1 Spring Model of Single-lap Joint
The spring model of single lap 2-bolt joint model is
shown in Fig.5.

Fig.5. Spring model of single-lap two-bolt joint
2
K and
4
K are the stiffness of the plate sections
between the two bolts,
1
K and
6
K are the stiffness of
the plate sections from the bolts to the ends of the
plates, which can be calculated by
l
EA
.
3
K and
5
K
Plate 2
Plate 1
Bolt
are the stiffness of the fasteners in the single-lap
joint, the reciprocal of the fasteners flexibilities
which can be calculated through equation(3).
1
K and
6
K just influence the stiffness of the whole
joint system and have little impact on the fasteners
load sharing, therefore, when
5
u is treated as zero,
the balance equations can be written as below:

= +
= +
= +
0 ) (
0 ) (
) ( ) (
4 4 2 4 3
3 5 2 3 2
4 2 3 3 2 2
u K u u K
u K u u K
P u u K u u K
(22)
Calculate
2
u ,
3
u and
4
u :
) (
5 2
2
2
4 3
2
3
3 2
2
K K
K
K K
K
K K
P
u
+

+
+
=
(23)
) )( (
5 2
2
2
4 3
2
3
3 2 5 2
2
3
K K
K
K K
K
K K K K
P K
u
+

+
+ +
=
(24)
) )( (
5 2
2
2
4 3
2
3
3 2 4 3
3
4
K K
K
K K
K
K K K K
P K
u
+

+
+ +
=
(25)
The loads carried by bolt 1 and bolt 2 are:
) (
5 3 5 1
u u K P = (26)
) (
4 2 3 2
u u K P = (27)
The displacement of the whole joint under load P ,
6 1
5 2
2
2
4 3
2
3
3 2
1
) (
K
P
K
P
K K
K
K K
K
K K
P
u + +
+

+
+
=
(28)
The load distribution of 3-bolt single-lap joint can be
obtain through the same method with that of 2-bolt
joint.
3.2 Model Validation
Load distribution tests following ASTM D 7248
standard (shown in Fig.7) were used to validate the
load distribution calculation method based on the
spring model mentioned above.
There are 3 groups of specimens, whose geometry
configuration of the joints are shown in table.1,
where t1, t2, W, e, p, D stand for the two plate
thicknesses, widths, edge distance, row distance and
hole diameter respectively. Table.3 listed the
equivalent engineering constants of the laminate
plate. The detail test method, parameters and results
can also refer to reference 7.

Fig.7 Load distribution test
Table.1 geometry configuration
Test
No.
W
(mm)
e
(mm)
P
(mm)
t1
(mm)
t2
(mm)
D
(mm)
1 36.47 15.17 29.83 3.81 3.81 5
2 36 20 25 2.5 7.5 5
3 60 36 60 10 32.57 12
Table.2 mechanical properties
Test No. E
1
(GPa) E
2
(GPa) G
12
(GPa)
12

1 131 8.8 5.2 0.3
2 131 8.11 3.66 0.34
3 135 9.4 5 0.28
Table.3 Lay-ups and equivalent constants
No. lay-up E
x
(GPa) E
y
(GPa) G
xy
(GPa)
xy
1
[45/0/-
45/90/0]
3s

63.2 40.7 13.5 0.364
2
[45/0/-45/0/90
/0/45/0/-45/0]
s

73.5 29.7 15.2 0.44
3
[45/0/-45/0/90
/0/45/0/-45/0]
4s

79.8 32.9 17 0.41
The load distribution of different bolts is calculated
using fastener flexibility equation from ASTM D
7248
[8]
(Equation 29), Hart-Smith
[9]
(Equation 30)
and this paper (Equation 21). The fastener flexibility
equations are shown as below. The results of the
load distribution are shown in Table.4, from which it
shows that the result calculated through equation
presented in this paper is more accurate than the
other two.
xP P xS S F P S
P S
F
P P S S
F
F P S
F
E t E t E t t
t t
d E
t t t t
d E
t t
C
2 1 2
192
) 16 8 ( 64
3
) 1 )( 2 ( 8
4
3 2 3
2
+ +
+
+
+ +
+
+ +
=

(29)
( ) ( )
) 3 1 )(
1 1
(
3
) ( 2 1
2
2
1
1 2 1
2 1
2 1

+ + +
+
+
+
= =
T L T L bbr
b b
E E t E E t E t t
t t
A G
t t
P K (30)

5
PAPER TITLE
Table.4 load distribution results (in percentage)
Test No.
ASTM
D 7248
Hart-
smith
Liu-Mao Test
1 52.72 53.76 54.65 56.23
Error (%) 6.2 4.4 2.8
2 54.93 54.16 56.92 57.22
Error (%) 4.0 5.3 0.5
3 51.51 53.42 56.68 57.83
Error (%) 11.0 7.6 2.0
3.3 Load Distribution Analytical Tool in Excel
The analytical tools, compiled in Excel, are shown
in Figure.8 and Figure.9, in which the parameters are
thickness of plate1 (t1) and plate2 (t2), distances
from the bolts to the plate ends (L1 and L2), elastic
modulus of the plates and bolts (E1, E2, E3 and E4),
plate width (W), row distance (p), and the total load
(P). Upon entering all the parameters, the calculation
will be accomplished automatically. Moreover, the
analytical tools could be used in the calculation with
bolts that have different diameters and elastic
modulus, hence a great convenience for the
engineers at the beginning of structure design.

Fig.8 the analytical tool for 2 fasteners

Fig.9 the analytical tool for 3 fasteners
4 Influences of Different Factors on Load
Distribution
4.1 Ratio of Bolt Diameter to Plate Thickness
For the single-lap joint, one of the plate is laminate,
with stacking sequence being [0/45/90]16S, ply
thickness being 0.125mm, and the material
properties is shown in Table.5; the other is 8mm
thick 30CrMnSi plate(E=196GPa,=0.3).
Table.5 Material properties of T300/QY8911
E
11
(GPa) E
22
(GPa) G
12
(Gpa)
12

135 8.8 4.47 0.33
The hole diameters distinguished by D=6mm, 8mm,
10mm and 12mm.The load distribution got from the
analytical tool is plotted in Fig.11.
45
46
47
48
49
50
51
52
6 8 10 12
Bolt Diameter(mm)
L
o
a
d

D
i
s
t
r
i
b
u
t
i
o
n
(
%
)
Bolt 1 Bolt 2

(a) 2 bolts
0
10
20
30
40
50
6 8 10 12
Bolt Diameter(mm)
L
o
a
d

D
i
s
t
r
i
b
u
t
i
o
n
(
%
)Bolt 1 Bolt 2 Bolt 3

(b) 3 bolts
Fig.11 Influence of t/d
Both the diagrams (a) and (b) indicate that the most
loaded bolt endures much more load, with the
increasing of the bolt diameter. The change of bolt
diameter causes the proportion of the stiffness
between bolt and plates which result in the uneven
load distribution should be the explanation.
4.2 Ratio of Row Distance to Plate Thickness
Use the same plates depicted in section 4.1, but the
row distance is 3t, 5t and 7t, respectively.
0
10
20
30
40
50
60
70
48 80 112
Row Distance(mm)
L
o
a
d

D
i
s
t
r
i
b
u
t
i
o
n
(
%
)Bolt 1 Bolt 2

(a) 2 bolts
0
10
20
30
40
50
60
48 80 112
Row Distance(mm)
L
o
a
d

D
i
s
t
r
i
b
u
t
i
o
n
(
%
)Bolt 1 Bolt 2 Bolt 3

(b) 3 bolts
Fig.12 Influence of p/t
The results in Fig.12 (a) imply the distribution
becomes more uneven among fasteners with the
increase of row distance.
4.3 Ratio of Plate Width to Plate Thickness
With all the other parameters identical with section
4.1, the variable s, column distance varies from 3D
to 5D. Results in the two-fastener and three- fastener
situations are plotted in Fig.13.
0
10
20
30
40
50
60
70
18 24 30
Column Distance(mm)
L
o
a
d

D
i
s
t
r
i
b
u
t
i
o
n
(
%
)
Bolt 1 Bolt 2

(a) 2 bolts
0
10
20
30
40
50
60
70
18 24 30
Column distance(mm)
L
o
a
d

D
i
s
t
r
i
b
u
t
i
o
n
(
%
)Bolt 1 Bolt 2 Bolt 3

(b) 3 bolts
Fig.13 Influence of W/t
As shown in Fig.13, the load distributions in
different bolts remain nearly the same with the
variation of column distance. These indicate that the
column distance is not as significant as other
parameters discussed above in the influence over
load distribution.
Conclusions
(1)The calculation tool getting load distributions of
thick-laminate multi-bolt single-lap joint was
validated to be accurate and efficient, especially
when dealing with the thick plates.
(2)Considering that in most of the situations, the
joint area shall be divided into several regions
containing the original number of rows but only one
column of fasteners, accuracy of the solution could
be guaranteed. Thus, it is a very useful tool for the
designers to commence the evaluation of parameters
at the very beginning.
(3)The increasing of bolt diameters and row
distances has impact on the load distribution, which
develops more uneven among fasteners. Due to the
simplified method which divides the multi-row,
multi-column joint area into multi-row, single-
column sub areas, the influence of column distance
over load distribution is equivalent of the influence
of plate width of the sub area over load distribution,
which has little effect on the load distribution.
(4) The stiffness ratio has great influence on the
load distribution, which differs more seriously
between fasteners when raising the stiffness ratio.
Besides, the distribution becomes more uneven
when the ratio of fasteners distance to the plate
thickness comes larger.
References
[1] M.A. McCarthy, C.T. McCarthy, G.S. Padhi A
simple method for determining the effects of
bolthole clearance on load distribution in
single-column multi-bolt composite joints.
Composite Structures, Vol. 73, pp 78-87, 2006.
[2] Roman Starikov Fatigue behaviour of
mechanically fastened aluminium joints tested in
spectrum loading. International Journal of
Fatigue, Vol. 26, pp 1115-1127, 2004.
[3] Johan Ekh, Joakin Schon Finite element
modeling and optimization of load transfer in
multi-fastener joints using Structural elements.
Composite Structures, Vol. 82, pp 245-256,
2008.
[4] B. Andersson Optimization and statistical
analysis of bolted joints-Part A: Theory and
system verification. Composite Science and
Technology, Vol. 66, pp 875-885, 2006.
[5] Timoshenko,S., Gere,J. Mechanics of
Materials, Science Press, Beijing, 1990
[6] D5961/D5961M-08. Standard Test Method for
Bearing Response of Polymer Matrix Composite
Laminates. Composite Materials. ASTM
International. West Conshohocken, 2010.
[7] Y.Mao Study of load distribution in multi-bolt
thick laminate joints, Thesis of Shanghai Jiao
Tong University,2011
[8] D 7248/D7248M08. Standard Test Method for
Bearing/Bypass Interaction Response of
Polymer Matrix Composite Laminates Using 2-
Fastener Specimens. Composite Materials.
ASTM International. West Conshohocken, 2010.
[9] Hsien-Yang Yeh, Johnathan J. Lee,Daniel Y. T.
Yang Study of multirow highly loaded bolt
joints in composite wing structure.Journal of
Aircraft,Vol.44(2),pp 380-385,2004