Modeling the Normal Contact Characteristics between Components
Joined in Multi-Bolted Systems
RAFAŁ GRZEJDA
Faculty of Mechanical Engineering and Mechatronics,
West Pomeranian University of Technology in Szczecin,
19 Piastow Ave., 70-310 Szczecin,
POLAND
Abstract: - This article is concerned with the modeling and calculation of the contact layer between components
joined in a multi-bolted system for assembly conditions. The physical model of the multi-bolted connection is
based on a system consisting of an elastic flange component, which is joined to an elastic support using a rigid-
body bolt model. The contact layer between the joined components is described by a non-linear Winkler model.
A model of the contact joint with consideration of the experimental normal elastic characteristics is presented.
Examples of normal contact pressure distributions are included.
Key-Words: - contact joint, normal characteristics, multi-bolted system, preload, bolt-tightening sequence,
finite element method.
Received: July 9, 2023. Revised: March 2, 2024. Accepted: April 11, 2024. Published: May 16, 2024.
1 Introduction
Multi-bolted systems are used for the mechanical
fastening of components in all fields of engineering,
particularly in mechanical or civil engineering, [1],
[2], [3], [4], [5], [6], [7], [8], [9]. At present, they
are more and more applied to join not only steel
components, but also composite, [10], [11], [12] or
additively manufactured components, [13], [14],
[15]. There are even times when polylactic acid
bolts manufactured by fused deposition modeling
are used, [16]. Multi-bolted systems are therefore
one of the most common detachable connections
used in engineering practice.
In practice, multi-bolted connections used in
preloaded systems are particularly important, [17],
[18], [19], [20], [21], [22]. With a properly
conducted preloading process, it is possible to
prevent both self-loosening of the connection, [23],
[24] and the phenomenon of connection unsealing,
[25]. In addition, the gradual tightening of bolts
during the assembly process, [26], [27], [28], [29]
and the subsequent loading of preloaded systems
with the working force, [30], [31] is equivalent to a
variable contact layer pressure between the joined
components. Hence, work on modeling the contact
layer between the components of such connections
is much needed.
Currently, finite element systems, [32], [33],
[34], [35] are often applied in the modeling of
contact joints in multi-bolted systems. Nevertheless,
unfortunately, using the standard contact elements
available in these systems, only constant stiffness
coefficients for each contact element at the contact
surface can be considered. In articles [31] and [36],
numerical analyzes were carried out to determine
the contact pressure at the interface between a
serrated or flat gasket and a flange in a pipe
connection. They used TARGET 170 and CONTA
174 contact elements available in ANSYS software
with standard settings. The same type of finite
elements and the same method for modeling contact
joints were used in [33], [37], [38], [39], [40] among
others. Authors using another popular commercial
FEM software, ABAQUS, often assume contact
properties as 'hard' contact in the normal direction
without friction [41] or with friction, [42], [43],
[44]. The same is true for another FEM software,
Midas NFX. In this case, also, constant normal and
tangential stiffness coefficients and a friction
coefficient are inserted, [45]. In some works, the
stiffness of the contact layer is not mentioned in
detail in the modeling. It is then stated that the
contact has been modelled in a standard way in the
relevant software, [46], [47], [48].
In more advanced research, for example in [49],
[50], [51], the contact layer between the components
to be bolted together is replaced by a so-called
virtual material, with which the features of the
rough surfaces in contact are defined. Such features
can be determined in experimental tests and can be
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specified using the mechanical characteristics of the
contact.
In the contact analysis of components joined in
a multi-bolted system with experimentally defined
characteristics, it is essential to take into account
changes in the stiffness coefficients for each contact
layer element. In such a case, further specific
computational procedures are required and
implemented in conjunction with the calculations
performed in the finite element system.
Model tests of preload variation in multi-bolted
connections can be validated by experimentally
monitoring the forces in the bolts. Methods used in
this regard are presented in [52], [53], [54] among
others.
2 Model of the Multi-Bolted System
The general structure of the multi-bolted system is
shown in Figure 1. The model consists of a pair of
flexible components joined by bolts. A non-linear
contact layer is introduced between the joined
components. These four parts of the connection are
regarded as subsystems of the multi-bolted system.
The subsystems are denoted by symbols according
to the following list:
• B – subsystem of bolts;
• F – joined component of the flange type;
• C – conventional non-linear contact layer;
• S – joined component of the support type.
a)
b)
Fig. 1: Multi-bolted system: a) scheme, b) model of
the system
By adopting the finite element method as the
modeling method, the system components (flange
and support) can be modeled using spatial finite
elements. Bolts, on the other hand, can be modeled
using a wide variety of models, including bar, beam,
and spatial models.
The non-linear contact layer between joined
components is modeled as a Winkler model, which
is defined by a set of j (j = 1, 2, ..., l) one-sided
springs, described by the following relation:
(1)
where: Rj – force in the center of the j-th elementary
contact area, Aj – j-th elementary contact area, unj –
normal deformation of the j-th non-linear spring,
respectively (for a review, see Figure 1).
The creation of the contact layer model proceeds
in the following steps:
1. Dividing the contact area between the joined
components (Figure 2a) into elementary contact
areas (Figure 2b).
2. Addition of mesh nodes at the centers of gravity
of the elementary contact areas (Figure 2b).
3. Insertion of non-linear springs at the nodes
identified in the previous step.
4. Creation of a 2D finite element mesh on the
contact area (Figure 2c).
Based on the 2D finite element mesh,
a homogeneous 3D finite element mesh is generated
at the interface between the joined components for
the entire volume of the joined components.
The equilibrium equation of the system for the
whole multi-bolted system can be written in the
form:
(2)
where: K – stiffness matrix, q – vector of
displacements, p – vector of loads, respectively.
Given the above division of the system into
subsystems, Eq. (2) can be presented as:
(3)
where: Ka – stiffness matrix of the a-th subsystem,
Kab – matrix of elastic couplings between the a-th
and b-th subsystems, qa – vector of displacements of
the a-th subsystem, pa – vector of loads of the a-th
subsystem, respectively, (a, b – symbols of the
subsystems, a {B, F, C, S}, b {B, F, C, S}).
By solving the system of equations, a columnar
displacement vector qC is obtained:
(4)
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In the next step, the forces Rj can be determined
by Eq. (1).
The solution of Eq. (3) is carried out in an
iterative process using the secant method, [55]. For
the tightening of the first bolt, the linearization
follows the method shown in Figure 3a, starting
from the origin of the coordinate system. On the
other hand, in the case of tightening the next bolt, it
starts from the operating points Wj corresponding to
the preload of the j-th non-linear spring in the
previous calculation step (Figure 3b).
a)
b)
c)
Fig. 2: Creating the finite element mesh at the
contact surface between the joined components: a)
real contact area, b) elementary contact areas, c)
scheme of the mesh
At each tightening stage of the multi-bolted
system, the linearization process is carried out until
the following condition is met:
(5)
where: R’c – reaction in the j-th non-linear spring
obtained from the linearization, Rc – actual reaction
in the j-th non-linear spring, c – index dependent on
the case of the calculation process (c {j, mj}),
–
permissible linearization error, respectively.
Based on the obtained values of the normal
deformation of the j-th non-linear spring unj, the
normal contact pressure pj on the j-th elementary
contact surface can be determined according to the
relation:
(6)
where: czj — stiffness coefficient of the j-th spring
model (for a review, see Figure 1b).
As the next bolt to be tightened, the bolt lying in
the area of the lowest mean normal pressure on the
contact surface of the joined components is selected.
a)
b)
Fig. 3: Linearization of a curve by the secant
method: a) in the case of the first bolt tightening, b)
in the case of the next bolt tightening
3 Numerical Example
According to the presented method, the multi-bolted
system shown in Figure 4a was calculated.
The calculations were carried out using the
Midas NFX 2023 R1 software, [56]. A rigid body
bolt model was used as the bolt model, [57], [58],
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[59]. In contrast, the components were modeled
using spatial elements, and the contact layer using
non-linear spring models. The connection was fixed
with seven M10 bolts. The calculations were
performed for a thickness of the connected
components h equal to 20 mm and a bolt preload Fmi
equal to 20 kN. The optimal bolt-tightening
sequence determined for the adopted multi-bolted
system is set out in Table 1. This sequence is similar
to a standard star pattern, [60], [61].
a)
b)
Fig. 4: FEM-based model of a multi-bolted system:
a) adopted bolt numbering, b) nodes adopted to
describe the distribution of normal contact pressure
Table 1. The optimal bolt-tightening sequence
Tightening
Sequence
1
2
3
4
5
6
7
Bolt number
1
4
6
2
5
7
3
The stiffness characteristics of the springs lying
in the contact layer were described by the following
experimentally determined function, [18]:
(7)
whereby the quantities occurring in this formula are
analogous to those in Eq. (1).
The numbering of the nodes used to describe the
normal contact pressure distribution is shown in
Figure 4b. The distribution of the normal contact
pressure on the component surfaces along the line
connecting the nodes indicated in Figure 4b, during
the preloading process of the system, is shown in
Figure 5.
Fig. 5: Diagrams of normal pressure during the preload process for individual bolts in the system
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
1 - Bolt No. 1
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
2 - Bolt No. 4
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
3 - Bolt No. 6
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
4 - Bolt No. 2
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
5 - Bolt No. 5
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
6 - Bolt No. 7
0
4
8
12
16
20
24
28
1 3 5 7 9 11 13 15 17 19
pn[MPa]
Number of the node
7 - Bolt No. 3
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An assessment of the final normal pressure
values can be made using the N-index:
(8)
where: pn — the value of the normal contact
pressure on the n-th contact surface, associated with
the n-th node (Figure 4b, n = 1, 2, …, 19); pan —
mean value of the normal contact pressure on the
line joining the nodes shown in Figure 4b,
respectively.
The values of the N-index range from -14.3 to
18.6%. This fact indicates a large variation in the
value of the normal contact pressure on the analyzed
surface of the joined components compared to their
average value. Authors of such articles as [62], [63],
[64], among others, came to similar conclusions
based on studies of the preload process of various
multi-bolted connections.
4 Conclusion
The presented model of a multi-bolted system can
be effectively used in the analysis of bolt preload
variation, once it has been prepared using CAx
techniques, [65], [66]. Its implementation allows the
control of the current value of the normal contact
pressure between joined components and enables
the selection of the optimal bolt-tightening
sequence, which is particularly important for thin-
walled structures used in aviation, [67], [68]. It is
planned to further develop the model to take into
account the operating condition of the multi-bolted
system.
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