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 rigidbody 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

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 CONTA174 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 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

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:

where:

– force in the center of the j-th elementary contact area

– j-th elementary contact area

–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:

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:

where:

– stiffness matrix of the a-th subsystem

– matrix of elastic couplings between the a-th and b-th subsystems

– vector of displacements of the a-th subsystem

– vector of loads of the a-th subsystem, respectively, (a, b – symbols of the subsystems,

By solving the system of equations, a columnar displacement vector is obtained:

In the next step, the forces 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).

At each tightening stage of the multi-bolted system, the linearization process is carried out until the following condition is met:

where:

– 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

α – permissible linearization error, respectively.

Based on the obtained values of the normal deformation of the j-th non-linear spring

, the normal contact pressure

on the j-th elementary contact surface can be determined according to the relation:

where:

— 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.

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],[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].

The stiffness characteristics of the springs lying in the contact layer were described by the following experimentally determined function, [18]:

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.

An assessment of the final normal pressure values can be made using the N-index:

where:

— 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)

—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 thinwalled 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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DOI: 10.37394/232011.2024.19.8

来源：midas机械事业部

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