Numerical Investigation of the Variability of Bolt Forces in a Preloaded

Asymmetric Multi-Bolted Connection under Cyclical Loading

RAFAŁ GRZEJDA

Faculty of Mechanical Engineering and Mechatronics,

West Pomeranian University of Technology in Szczecin,

19 Piastow Ave., 70-310 Szczecin,

POLAND

Abstract: - A numerical study of a seven-bolt connection with an asymmetric contact surface between the

components to be joined is reported. The investigations were organised into two steps. Firstly, the connection

was preloaded in a three-pass cycle. Then, the connection was subjected to the cyclically varying force imposed

at an angle of 30 degrees to the joined components' contact surfaces to produce both compressive and shear

loads in the connection. The connection modelling was performed in the finite element method convention. The

joined components were discretized using three-dimensional finite elements and the fasteners were modelled as

special elements consisting of flexible beams, stiff heads, and stiff nuts. The article is concluded by analysing

selected computational outcomes.

Key-Words: - multi-bolted connection, preload, cyclical loading, FE analysis

Received: January 22, 2023. Revised: April 14, 2023. Accepted: May 11, 2023. Published: June 22, 2023.

1 Introduction

Steel and aluminum multi-bolted connections with

preloaded bolts are one of the most widely applied

nodes in various engineering fields. Their proper

functioning depends mainly on the behaviour of the

bolt forces. With cyclical loads applied to both

single-bolted connections and multi-bolted

connections, the variability of these forces can be

noticed, [1], [2], [3], [4]. Lack of adequate

mechanical protection may even lead to self-

loosening of the connections, [5], [6]. An overview

of the reasons and mechanisms of rotational and

non-rotational bolt loosening was reported by [7].

Actions to prevent loosening of multi-bolted

connections under dynamic loads include, but are

not limited to, coating the bolts, [8]. Furthermore,

the cyclical tensile behaviour of preloaded bolts

made in class 10.9, widely adopted in engineering

structures, was discussed by [9].

Testing of cyclically loaded multi-bolted

connections usually consists in observing the

deformation of the components of these connections

during their operation. It leads to the determination

of hysteretic stiffness curves and the formulation of

conclusions regarding the considered connection.

Selected numerical studies on this subject include

the following articles. In [10], the authors depicted

the results of testing the behaviour of a semi-rigid

frame specimen loaded cyclically. In [11], the

authors analysed a hollow section beam-to-column

connection in order to determine their ultimate

behaviour. In [12], the authors described the

research of the innovative double-split tee (DST)

connection subjected to cyclical loading. Based on

the research results, they have demonstrated the

possibility of using DST connections with friction

shock absorbers in structures subjected to seismic

action. In [13], the authors observed the behaviour

of extended end-plate bolted connections under

monotonical and cyclical loads. They noted that

cyclical loading reduces both the bending and

rotational resistance of such connections.

Testing of cyclically loaded multi-bolted

connections less frequently relies on the observation

of changes in bolt forces during the operation of the

connections. In [14], the authors determined the bolt

preload degeneration in the reinforced legs of

a retrofitted transmission tower under cyclical

loading. They performed experimental tests on

a section of the selected leg and then validated the

results using a simplified model consisting of non-

linear springs and rigid links, created using the finite

element method (FEM), [15], [16]. In [17], the

authors investigated the variability of the bolt

preload in a double-shear lap connection under

cyclical loads. They carried out the analysis for half

of the connection model, due to its symmetry, and

using three-dimensional finite elements. In [18], the

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Rafał Grzejda

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author predicted the operating life of a gasket in a

bolted flange connection under cyclical bending,

while observing changes in forces in the bolts. He

conducted the analysis for a full flange connection

model built from three-dimensional finite elements.

There are also articles focusing on the

observation of bolt failure mechanisms in cyclically

loaded bolted connections, [19], [20].

The publications cited above are concerned with

typical multi-bolted connections, usually

symmetrical, or single-bolted connections. So far,

less attention has been paid to multi-bolted

connections with asymmetrical bolt arrangements,

[21], [22]. This gap is filled by the present article,

which analyses an asymmetric connection loaded

cyclically by forces inducing both normal and

tangential stresses in the connection. The objective

set in the article is to investigate the variability of

bolt forces in such a loaded connection using finite

element modelling. An analysis of the variability of

bolt forces in the multi-bolted connection was

carried out both at the preloading step of the

connection in a three-pass cycle and after loading

with a cyclically varying external force. It was

shown that, as a result of applying this force, the

bolt forces also change in a cyclical manner, and

that this variability depends on the position of the

bolt in the asymmetrical bolt arrangement in the

connection. This has not yet been shown in other

papers, as demonstrated above.

2 Problem Formulation

The multi-bolted connection investigated in this

study is illustrated in Figure 1 and consists of a pair

of components connected by i fasteners with

M101.25 threads (i



{1, 2, 3, …, 7}).

Fig. 1: Details of the physical multi-bolted

connection model

Each component to be joined in a connection

consists of two plates. One of them is inclined and is

the main connection plate while the other, referred

to as the base plate, is located in the XOY plane

(indicated in Figure 1). The components were

formed from 1.0577 steel, while the bolts were

made in class 8.8 and the nuts in class 8. In order to

minimise the number of contact joints, no washers

were used in the connection. The contact surface of

the joined main plates is angled at 60 degrees from

the horizontal so that the connection can be

subjected to compressive and shear loads

simultaneously, [23]. The contact surface shape

between the joined main plates is asymmetrical.

As demonstrated in the first section of the article,

FEM is at present the most common way for

modelling multi-bolted connections. In the models

developed using this method, the components to be

connected are generally modelled using three-

dimensional elements, [24], [25]. Fasteners, on the

other hand, are modelled in several different ways,

which are mentioned in [26], among others. After

assessing the existing bolt models in the literature,

in this article, a flexible beam model with a stiff

head and a stiff nut was chosen to model the

fasteners, [27], [28]. A multi-bolted connection

model structured in the convention of the finite

element method is shown in Figure 2.

Fig. 2: Details of the discrete multi-bolted

connection model

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All parts of the connection were attributed to the

properties of linear isotropic steel materials. The

constitutive relations, in this case, can be

characterised by Hooke's law, [29]. The constants

for the materials used in the discrete connection

model, including Elastic modulus E and Poisson's

ratio n, are summarised in Table 1. They correspond

to the steels with the characteristics given at the

beginning of Section 2.

Table 1. Characteristics of the materials used in the

discrete multi-bolted connection model.

Components

E, GPa



Fasteners

210

0.28

Main plates

210

0.3

Base plates

210

0.3

‘Welded’ contact elements were inserted

between the main connection plates and the base

plates to avoid them moving in any direction

relative to each other (Figure 2), in accordance with

the real implementation of the connection. ‘Welded’

contact elements were also applied between the

main connection plates. This procedure in numerical

simulations is common practice for preloaded

connections, [30], [31], [32].

The FE-based multi-bolted connection model

was generated with a total of 93,283 elements and

157,559 nodes. The maximum side dimension of the

finite element in the mesh is less than 10 mm. The

mesh was considerably thickened at the interface

between the main plates and at the interface between

the fasteners and the main plates (compare with

[33]).

The connection was fully restrained at all nodes

on the bottom surface of the lower base plate.

The test procedure was organised into two steps.

Firstly, the multi-bolted connection was preloaded

in accordance with the tightening method and order

established experimentally beforehand as the most

favourable with regard to the end distribution of bolt

forces after the tightening process, [34]. The bolt

preload value Fp0 was set to be equal to 22 kN based

on an analysis of the allowable pressure values

between the nuts and the lower main plate in the

multi-bolted connection. The connection was

preloaded in a three-pass cycle, sequentially

tightening bolts with numbers: 1, 4, 7, 3, 6, 2, and 5

(the adopted bolt numbering is provided in Figure

1). In the first pass, the bolts were preloaded to

0.2∙Fp0, in the second pass to 0.6∙Fp0, and in the third

pass to Fp0.

In the second step, the preloaded connection was

subjected to an external Fe cyclical force, the

variability of which is depicted in Figure 3. The

direction of this load changed repeatedly from top to

bottom and vice versa (for comparison see, [13]).

Fig. 3: Operating load variability

The external load was imposed in the Z direction

uniformly across the top surface of the upper base

plate. The maximum for the Fe force was chosen so

that the shear loads it induced were less than the

frictional forces acting at the contact of the joined

main plates.

3 Problem Solution

The calculations were performed in Midas NFX

2020 R2 using a sequential non-linear module.

Fig. 4: Bolt force distribution during connection

preloading

-40

-30

-20

-10

010 20 30 40

Fe , kN

t, s

0.99

0.992

0.994

0.996

Fpi / Fp0

0.2

0.4

0.6

0.8

Fpi / Fp0

Bolt No. 1 Bolt No. 2 Bolt No. 3 Bolt No. 4

Bolt No. 5 Bolt No. 6 Bolt No. 7

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The module includes a non-linear static analysis

for the bolt preloading phase as well as a non-linear

implicit transient analysis, [35], for the connection

operating phase.

The distribution of forces in the bolts during the

preloading of the connection is shown in Figure 4.

The graph shows the unified values of the bolt

preload forces Fpi related to the base value Fp0.

The values of the bolt forces after preloading

decrease as the tightening of the connection

progress. A quantitative assessment of this

variability can be made based on the W1 indicator

given by the formula:

𝑊

1=𝐹

𝑝0−𝐹𝑝𝑖

𝐹

𝑝0

(1)

Fig. 5: Bolt force distributions during operational

loading of the connection

0.996

0.998

1.002

1.004

010 20 30 40

Fb1 / Fp71

t, s

Bolt No. 1

0.996

0.998

1.002

1.004

010 20 30 40

Fb2 / Fp72

t, s

Bolt No. 2

0.996

0.998

1.002

1.004

010 20 30 40

Fb3 / Fp73

t, s

Bolt No. 3

0.996

0.998

1.002

1.004

010 20 30 40

Fb4 / Fp74

t, s

Bolt No. 4

0.996

0.998

1.002

1.004

010 20 30 40

Fb5 / Fp75

t, s

Bolt No. 5

0.996

0.998

1.002

1.004

010 20 30 40

Fb6 / Fp76

t, s

Bolt No. 6

0.996

0.998

1.002

1.004

010 20 30 40

Fb7 / Fp77

t, s

Bolt No. 7

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The values of the W1 indicator as a function of

bolt number i are listed in Table 2.

Table 2. W1 indicator values.

W1, %



0.68

0.80

0.74

0.65

0.84

0.73

The noticeable only slight reduction in force in

a particular bolt under preloading of subsequent

bolts is due to the adoption of a rigid type of contact

joint (i.e. ‘welded’ contact) between the main

connection plates. The negligible discrepancy in

bolt preload values in relation to its base value is

also influenced by the way the tightening process is

carried out, i.e. its implementation in three passes.

The bolt force distributions in relation to the

operating load obtained from the calculations are

shown in Figure 5. The graphs show the unified

values of the bolt forces Fbi related to the bolt forces

at the end of the tightening process Fp7i.

The values of the bolt forces after external

loading change as the loading of the connection

progress. A quantitative assessment of this

variability can be made on the basis of the W2

indicator given by the formula:

𝑊

2=𝐹𝑝7𝑖−𝐹𝑏𝑖

𝐹𝑝7𝑖

(2)

The values of the W2 indicator as a function of

bolt number i are listed in Table 3.

Table 3. W2 indicator values.

W2, %



0.05

-0.12

0.40

0.22

0.27

0.39

-0.03

The forces in the bolts located above the centre

of gravity of the contact surface between the main

connection plates (i.e. with numbers 3, 4, 5, and 6)

show more variability than for the other bolts. It is

also noticeable that not in all bolts the force changes

analogously to the operational force. In the case of

bolts numbered 2 and 7, these changes occur in the

opposite phase. However, the observed variations in

bolt forces are only minor and do not cause a loss of

load-bearing capacity of the multi-bolted

connection.

4 Conclusion

This article presents a numerical investigation of an

asymmetric multi-bolted connection successively

preloaded and then cyclically exposed to normal and

tangential loads. The following findings can be

extracted from the test outcomes:

1. Conducting the tightening process of a multi-

bolted connection in a series of passes results in

a relatively uniform distribution of preload force

in the bolts at the process end.

2. The forces in the individual bolts vary due to the

alternating external load on the multi-bolted

connection. In the considered case, the decrease

in bolt forces at the end of the operating process

is only minor and does not cause a loss of load-

bearing capacity of the multi-bolted connection.

3. For preloaded multi-bolted connections, which

are loaded operationally with cyclical force, it is

advisable to model the contact between the

components to be joined as a rigid type of

contact (for example, as the ‘welded’ contact

proposed in Midas NFX 2020 R2). This

simplification significantly reduces the numerical

calculation time.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflicts of interest to declare.

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