Determination of Bolt Forces and Normal Contact Pressure Between Elements in the System With Many Bolts for its Assembly Conditions

The paper put forward a thesis that modelling dissymmetric non-linear multi-bolted connections as a system is possible. Modelling of the systems composed of four subsystems on the assembly state was presented. These subsystems included: a couple of joined elements (a flange and a support), a contact layer between them, and bolts. The physical model of the system was described considering the tightening of bolts according to a specific sequence. In this model: the flange and the support were built using spatial finite elements, the contact layer was formed as the non-linear Winkler model, and the bolts were replaced with simplified models made of flexible beams. The calculation model which can be applied to determine the changes in bolt forces, as well as in the normal contact pressure between the joined elements during the tightening of the system and at its end, was presented. The results of sample calculations for the selected multi-bolted system were shown.


INTRODUCTION
Multi-bolted connections are frequently applied in mechanical and civil engineering.Most often, they can occur in two states of loading and deformation.The first one is the assembly state in which the connection may be set up under different assembly patterns [2,32].Another feature of multi-bolted connections is that they are constructed of many structural components [7,22,27,28].Due to irregular geometrical structures on the contact surfaces of the joined elements, these connections are usually treated as non-linear [33].The subject of this paper is modelling of multi-bolted connections regarded as a non-linear system for the assembly condition.
In all the afore-mentioned publications, a systemic approach to modelling, calculation and analysing multi-bolted connections was not undertaken.
The FEM was also used in this paper for modelling and calculations of the multi-bolted system.The presented paper constitutes a continuation of the previous publication on modelling non-linear multi-bolted systems [14] in which the distribution of preload in bolts were analysed in the system model with the rigid body bolt models.The current paper concerns the analysis pertaining to preload of bolts distributions and distributions of normal contact pressure between the joined elements in the system model with the spider bolt models.

MODEL OF THE MULTI-BOLTED SYSTEM
The concept of system approach to modelling multi-bolted connections was presented earlier at the 3 rd Polish Congress of Mechanics [13].
The model was built from the four subsystems (Fig. 1 The spider bolt model was used as a single bolt model in the system; the plain part of the bolt and its head were modelled with use of beam elements but the total volume of the beam elements for the head was assumed to be equal to the volume of the head of the real bolt.
Generation of the contact layer model begins with the division of the contact surface between the flange and the support into elementary contact areas.Then, the following are added to the model: • mesh nodes in the centres of gravity of the elementary contact areas, • non-linear springs at the nodes identified in the previous step, • 2D finite element mesh on the contact surface considering the position of the non-linear springs.
The presented method enables the modelling of the contact layer with the characteristics obtained experimentally separately for each element of this layer.
On the basis of the 2D finite element mesh on the contact surface between the flange and the support, a uniform 3D finite element mesh was created for the entire volume of the joined elements.
After taking into consideration the division of the multi-bolted system into subsystems, the equation of system equilibrium can be presented in the form [13]: where: K XX denotes the stiffness matrix of the X-th subsystem, K XY is the matrix of elastic couplings between X-th and Y-th subsystems, q X is the vector of displacements of the X-th subsystem, and p X denotes the vector of loads of the X-th subsystem, (X and Y are the symbols of the subsystems, X = Y = {B, F, C, S}).

CALCULATIONS OF THE MULTI-BOLTED SYSTEM ON THE ASSEMBLY STATE
Sample calculations were performed for a selected asymmetrical multi-bolted system shown in Figure 2a.
The model was built using the Midas NFX 2017 program.The connection is fastened using seven M10 bolts made in the class of mechanical properties 8.8.The physical properties of the alloy steel used in the connection model for the bolts and the joined elements are shown in Table 1.
Calculations were performed for the joined elements with a thickness of 20 mm.The outer contours of the flange element and the support can be placed inside a rectangle with sides 110 mm and 125 mm.
The model of the system was fastened in the manner shown in Figure 2a by receiving: • all degrees of freedom on the bottom surface of the support, • degrees of freedom in a plane perpendicular to the axis of the bolts in five nodes lying on the outline of the upper surfaces of the flange and the support.
The preload of bolts was set at 20 kN.The tightening sequence is given in parentheses in Figure 2a.
Between the joined elements, a Winkler type of the contact layer built from a set of nonlinear springs was introduced.The stiffness characteristics of the springs are described by the following experimentally determined function [12]: where: R j denotes the force in the centre of the j-th elementary contact area, A j is the j-th elementary contact area, and u j denotes the deformation of the j-th non-linear spring of the contact layer.
As a result of the calculations, preload of bolts distributions and distributions of normal contact pressure between the joined elements during the tightening process and at the end of this process were obtained.
In Figure 3  In contrast, the diagrams of normal contact pressure on elementary surfaces on the line joining the nodes defined in Figure 2b during the tightening process of the multi-bolted system are shown in Figure 4.
As the last graph, the scatter of the final preload distribution in the bolts at the end of the tightening process is presented in Figure 5.
On the basis of the obtained results it can be seen that the distributions of the preload of the bolts as well as normal contact pressure between    the joined elements during the tightening process and at the end of this process are highly variable and irregular.
The assessment of final values of the preload of the bolts and normal contact pressure can be executed using the following indicators expressed as a percentage: [ [ where: F fin,i denotes the final value of the preload of the i-th bolt at the end of the tightening process, F mi is the initial value of the preload of the i-th bolt at the beginning of the tightening process, p n is the value of normal contact pressure on the n-th contact surface, linked to the n-th node (Fig. 2b, n = 1, 2, 3, …, 19), and p n,a denotes the average value of normal contact pressure on the line joining the nodes shown in Figure 2b.
According to the introduced multi-bolted system model, the preload values of bolts may vary between -10.8 and 0.2%, relative to their initial values.At the same time, the values of normal contact pressure between the joined elements may vary in the range of -16.6 to 33.7%, in relation to their average values.The resulting calculation procedures can be used to optimise the preload of the individual bolts in order to ensure uniform pressure distribution at the interface between elements connected in the multibolted system.This can be accomplished by examining the impact of the tightening sequence of bolts on distribution of the preloads and normal contact pressure between the joined elements.

CONCLUSIONS
The described model of the multi-bolted system can be successfully used in the analysis of preload variations and normal contact pressure between joined elements variations in the case of any connection of flexible elements.The model can also provide the analysis on how does the tightening sequence affect the preload values in bolts before the preloaded connection is loaded by an external force.
variations of the preload in individual bolts during the tightening process of the multi-bolted system are shown as follows: • in the first line -the preload distribution in bolts No. 1 and No. 4, which were tightened as the first and as the second bolt, respectively, • in the second line -the preload distribution in bolts No. 6 and No. 2, which were tightened as the third and the fourth bolt, respectively, • in the third line -the preload distribution in bolts No. 5, No. 7 and No. 3, which were tightened as the fifth, the sixth and the seventh bolt, respectively.

Fig. 2 .
Fig. 2. Analysed multi-bolted system: a) FEM-based model with the numbering of the bolts and the tightening sequence given in parentheses, b) nodes on the contact surface of the joined elements adopted to describe normal contact pressure distributions.

Fig. 3 .
Fig. 3. Preload distributions in the bolts during the tightening process of the multi-bolted system.

Fig. 4 .
Fig. 4. Diagrams of normal contact pressure on the contact surface of the joined elements during the tightening process of the multi-bolted system.

Table 1 .
Material properties of the steel used for the bolts and the joined elements in the connection model