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**For up-to-date bolted joints, first of all in vehicles, high strength bolts of 10.9 or even 12.9 are used, which are pre-tightened up to 90% or even 100% of the yield strength. The primary aim of this high degree utilization is the weight reduction. For the analytic dimensioning of bolted joints, the VDI 2230 Richtlinien German standard provides support. However, the analytic model can mostly consider the true structural characteristics only in a limited way. The analytic modeling is especially uncertain in case of multiple bolted joints when the load distribution among the bolts depends reasonably upon the elastic deformation of the participating elements in the joints over the geometry of the bolted joint. The first part of this paper deals with the problems of numerical modeling and stress analysis, respectively specifying the analytic dimensioning procedure by applying elastic or rather elastic-plastic material law. The error magnitude in bolted joint calculation was examined in case of omitting the existing threaded connection—between the bolt and the nut—in order to simplify the model. The second part of the paper deals with the dimensioning of stands and cantilevers’ multi-bolt fixing problems, first of all, with the load distribution among the bolts keeping in view the analysis of the local slipping relations. For demonstrating the above technique, an elaborated numeric procedure is presented for a four-bolted cantilever, having bolted joints pre-tightened to the yield strength.**

The safe operation of industrial equipments is principally determined by the applied bolted joints which are pretightened up to 90% or even 100% of the material yield strength. As a consequence, the bolt, the nut and the elements encircled between (in case washers as well) take the pre-load (stress) according to the force system equilibrium. In the joining elements, this force equilibrium determines the stress and the deformation states mostly in the elastic but in cases of the plastic range.

The related references introduce the bolted joint according to numerous aspects (ex. [

In the first part of the study, a finite element model was created applicable for calculating the pre-tightening and loosening of bolted joints. By this model, bolted joint mechanic behaviors were studied in cases of different geometry and loading conditions. The numeric results were comparable to the analytic calculations, and useful conclusions could be drawn relating to certain calculation models.

In this paper one M20 and one M24 bolted joints were studied. The geometric model of the bolted joints is shown in

threads are shown in

For M20 and M24 bolts the pre-tightening forces and the tightening torques were determined for 100% yield strength condition of the bolts in accordance with the recommendations of the VDI 2230 Richtlinien [

The behavior of bolted joint used to be analyzed in force-deformation diagram. With the knowledge of the

geometric and mechanic characteristics of bolted joint’ elements the data for constructing the force-deformation diagram were determined. Based on the data summarized in

The geometric model used for the numeric analysis is shown in

The connection between the bolt and the nut was studied in case of real threaded or simplified bonded connection cases. In case of threaded joint the FE model resulted large element numbers. In the case of bonded model (there is a bonded connection between the cylindrical surfaces of the bolt and the nut) this solution results reasonably reduced element numbers because of ignoring the real frictional contact behaviour.

The mechanical behavior of the highly loaded bolted joints may be influenced by the possible yielding of the structural elements included in the joint. Therefore calculations were carried out with linear elastic material law and with non-linear material law as well.

In case of M20 bolted joint, the compressed plate is a 65 mm cylindrical body having a 21 mm hole in the mid-

dle and for M24 bolt the compressed plate is a 120 mm cylindrical body having a 25 mm hole in the middle of the body. The dimensions, the FE model for its quarter, the loading and the boundary conditions are shown in

Main characteristics of the finite element model are:

• type of the applied element: tetra 10;

• average element size: 2 mm;

• average element size in the threaded connection: 0.4 mm;

• average element size in the plate-bolt connection: 0.7 mm;

• number of elements: 193,009;

• number of nodes: 288,660.

The bolts are pre-tightened up to the yield strength, consequently the loading of the quarter model is 64 kN, which force applied on the bolt model upward and downwards on the surfaces (

The coefficient of friction on the threads is μ = 0.1. It should be noted that the shearing loading resulted by the tightening torque of the nut is not considered by the model.

Utilizing the symmetric characteristics of the model the displacements of side surfaces¾perpendicular to the

surface¾must be prevented. This is ensured by fixing the edge of the plate bigger diameter of the model (in spatial position, see in

The z direction displacement field is shown in

The elastic deformations were determined by the displacement averages of the nodes on surfaces 1 and 2 and on surfaces 3 and 4 (

In

When the stress image is cut off by the yield strength of the washer (420 MPa) those zones can be seen by red, where the washer is already in the elastic-plastic zone (

By getting down (querying) the displacements it is possible to determine the sum of the bolt elastic elongation and the encircled elements compression and it makes possible to construct the pre-tightening force-deformation diagram (

sults. It can be seen that the threaded model reflects well the analytic result.

Similarly to the earlier analysis the calculation related to M24 bolted joint was executed too. The results obtained from the calculation provide the force-deformation diagram shown in

The bonded model differs from the model examined—in the earlier paragraph—in such a way that the threaded connection is not formed between the nut and the bolt. In the place of the threaded joint cylindrical surfaces are

formed having the diameter equal to the mean thread diameter and defining bonded joint between the two parts. The aim of this study is to estimate the resulted error magnitude of this approximation yielding to a smaller FE model, with less elements (This model takes into consideration the elastic deformation of the bolt shank and the compressed elements, but neglects the elastic behavior of the thread surroundings).

The geometry, the dimensions, FE quarter model, the loading and the boundary conditions for M20 bolted joint are shown in

Main characteristics of the finite element model:

• type of the applied element: tetra 10;

• average element size: 2 mm;

• average element size in the threaded connection: 0.4 mm;

• average element size in the plate-bolt connection: 0.7 mm;

• number of elements: 114,752;

• number of nodes: 170,660.

The loading of the model and the boundary conditions are the same as given in the previous paragraph.

The calculated z direction displacement field is shown in

In

The bonded model force-deformation diagram is shown in green color in

Similarly to the earlier study the related calculations were carried out for M24 bolted joints too. The calculation resulted force-deformation diagram is shown in

Based on the already completed studies it can be stated that the results of threaded finite element model answers to bolt dimensioning procedure of the VDI 2230 Richtlinien. The simplified bonded model increases the rigidity of the joint by 9%. The differences in the rigidity of the two models are resulted by the limited expansion¾in radial direction¾of the bonded joint, as the contact behavior between the threads are not considered here (

The studies carried out with linear material law¾in the previous two paragraphs¾pointed out that the stresses in the bolted joints reach and even exceed the yield strength in more components of the bolted joints. This made necessary the non-linear analysis. The concerned numerical model is shown in

In order to reduce the CPU time¾utilizing the approach of circular symmetry¾1/24-th of the model was examined (studied). The characteristics of the model are:

• applied element type: tetra 10;

• average element size: 2 mm;

• average element size at the thread: 0.4 mm;

• average element size in the plate-bolt connection: 0.7 mm;

• number of elements: 32,557;

• number of nodes: 54,628.

The bolt is pre-stressed¾up to the yield strength¾and 10.3 kN answers to the loading for the 1/24 (size) of the model.

The boundary conditions coincide to the boundary conditions written in the previous paragraph.

Over the yield strength the real behavior of the material applied is not known therefore the study was carried out with assumed material law. During the intake of the material law the gradient of the hardening part is determined in such a way that the stress level should reach the

ultimate strength at the tensile break strain.

Besides, the transition was made continuous between the elastic and plastic sections. The description of the assumed material law is summarized in

The calculated equivalent strains are illustrated in

in the range of 0 - 0.004. The strains were drawn in the deformed form of the joints by using deformation scale 1:1. Under the bolt head and washer substantial magnitude of deformation of the washer and the encircled elements can be seen well in the figures.

From z directional displacements the sum of the bolt elongation and compression of the encircled elements:

This considerable magnitude of elongation can be explained by the applied structural elements having low yield strengths.

The force-deformation diagram of bolted joint calculated with modified material law is shown in

In

The calculations were carried out for the maximum pre-tightening force (F_{M max}), which was obtained at the minimum friction coefficient and at the maximum tightening torque. However the probability is rather rare for these conditions. The non-linear calculations were car-

ried out for minimum (F_{Mmin}), and average (F_{Mk}) pretightening forces as well. These results are shown in

In the first part of this paper, the pre-tightening behavior of the high strength (10.9 és 12.9) bolted joints was summarized. At first, the VDI Richtlinien calculation method and then the finite element analysis were used. The FE model is also applicable for analyzing the contact state of the threaded contact showing a good conformity to the calculations according to VDI. The simplified¾so called bonded¾model shows 9% more rigid behavior due to the simplified surrounding contact.

The FEA model¾analyzing the elastic-plastic behavior¾demonstrates the smaller extent plastic behavior effect of the cantilever, the frame structure and also the washer. The bigger deformations obtained are the consequence of the smaller yield strength compared to the 90% - 100% pre-tightened bolt shank. Above all, it can be stated that during the stress analysis of pre-stressed high strength bolted joints, the expected elastic-plastic defor-

mations of the same elements of the joints must be considered, too.