Modelling Nonlinear Multi-Bolted Systems on the Assembly State

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Procedia Engineering 206 (2017) 1808–1812

International Conference on Industrial Engineering, ICIE 2017 International Conference on Industrial Engineering, ICIE 2017

Modelling Nonlinear Multi-Bolted Systems on the Assembly State Modelling Nonlinear Multi-Bolted Systems on the Assembly State R. Grzejda* R. Grzejda* West Pomeranian University of Technology, 19, Piastow Ave., Szczecin 70-310, Poland West Pomeranian University of Technology, 19, Piastow Ave., Szczecin 70-310, Poland

Abstract Abstract Modelling and calculations of asymmetrical nonlinear multi-bolted connections treated as a system composed of four subsystems are presented. subsystems are: a couple of joined elements (aconnections flange and treated a support), contactcomposed layer between elements, Modelling andThese calculations of asymmetrical nonlinear multi-bolted as a asystem of fourthesubsystems and a set of bolts. The physical model of the system on the assembly state is described taking into account the bolt preloading are presented. These subsystems are: a couple of joined elements (a flange and a support), a contact layer between the elements, conducted to physical a specificmodel sequence. the modelling the finitestate element method taking is used.into Theaccount flange and the support are and a set ofaccording bolts. The of theFor system on the assembly is described the bolt preloading built with spatial finite The contact is formed the as afinite nonlinear Winkler model, andThe theflange set of and boltsthearesupport modelled conducted according to aelements. specific sequence. Forlayer the modelling element method is used. are using simplified models (named as rigid body bolt models). The calculation presented, means which built with spatial beam finite elements. The contact layer is formed as a nonlinear Winkler model model, isand the set ofbybolts are of modelled changing force values the bolts duringasthe preloading of the systemThe and calculation at its end can be determined. Results of calculations using simplified beaminmodels (named rigid body bolt models). model is presented, by means of which for the exemplary multi-bolted system are the presented. changing force values in the bolts during preloading of the system and at its end can be determined. Results of calculations © 2017 The Authors. Published by Elsevier B.V. for the exemplary multi-bolted are presented. © 2017 The Authors. Publishedsystem by Ltd. committee of the International Conference on Industrial Engineering. Peer-review under responsibility of Elsevier the scientific © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering Keywords: multi-bolted connection; systemic approach;committee assembly state. Peer-review under responsibility of the scientific of the International Conference on Industrial Engineering. Keywords: multi-bolted connection; systemic approach; assembly state.

1. Introduction 1. Introduction Multi-bolted connections are often used in mechanical engineering, also for applications in industrial engineering. As itconnections has been indicated in [1], connections are characterized by the fact that they can Multi-bolted are often usedmulti-bolted in mechanical engineering, also for applications in industrial occur in two states of loading and deformation. The first of them is the assembly state. The bolts in this state cancan be engineering. As it has been indicated in [1], multi-bolted connections are characterized by the fact that they preloaded according to different assembly patterns [2-4]. The assembly state considered in relation to the multioccur in two states of loading and deformation. The first of them is the assembly state. The bolts in this state can be bolted connections as a nonlinear system is the[2-4]. subject this study.state considered in relation to the multipreloaded accordingtreated to different assembly patterns Theofassembly The source of the nonlinearity of multi-bolted connections are all of contact connections existing between joined bolted connections treated as a nonlinear system is the subject of this study. parts. In fact, such contact connections cause the geometrical nonlinearity in the system due the surface texture of The source of the nonlinearity of multi-bolted connections are all of contact connections existing between joined joined elements (for a review, see [5]). parts. In fact, such contact connections cause the geometrical nonlinearity in the system due the surface texture of joined elements (for a review, see [5]). * Corresponding author. Tel.: +48-91-449-4969; fax: +48-91-449-4564. E-mail address:author. [email protected] * Corresponding Tel.: +48-91-449-4969; fax: +48-91-449-4564.

E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier B.V. Peer-review the scientific committee 1877-7058 ©under 2017responsibility The Authors. of Published by Elsevier B.V.of the International Conference on Industrial Engineering . Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering .

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on Industrial Engineering. 10.1016/j.proeng.2017.10.717

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Papers on modelling and calculations of multi-bolted connections are usually focused on conventional types of joints such as:  beam-to-column connections [6-8],  lap connections [9-11],  flange connections [2,4,12]. In all the just mentioned publications, a systemic approach to modelling, calculations and analysing multi-bolted connections is not taken into account. The most popular method of modelling multi-bolted connections is the finite element method (FEM). While the joined elements in such connections are generally created as a spatial body, the bolts are modelled in different ways. Apart from spatial models of the bolts [2,4,6-8,12,13], the following substitute bolt FE-models are applied:  spring models [14,15],  rigid body bolt models with a flexible shank of the bolt and a rigid bolt head [16,17],  flexible beam bolt models [18,19]. In view of the above-mentioned fact, the FEM is also used in this paper for modelling and calculations of the multi-bolted system on the assembly state, and the rigid body bolt model as a bolt model is selected. 2. Model of the multi-bolted system The structure of the multi-bolted system model derives from the concept described in [20]. The model is built from the following four subsystems (Fig. 1):  B – the bolts,  F – the flexible flange element,  C – the conventional contact layer,  S – the flexible support. The equation of system equilibrium for the multi-bolted system can be presented in the form:

K q  p

(1)

where K is the stiffness matrix, q is the vector of displacements, and p is the vector of loads. After taking into consideration the division of the system into subsystems, Eq. (1) can be converted to the form:

 K BB K  FB  0   K SB

K BF

0

K FF K CF

K FC K CC

0

K SC

K BS   q B   p B  0  q F   p F    K CS   q C   pC       K SS   q S   p S 

(2)

where Kaa is the stiffness matrix of the a-th subsystem, Kab is the matrix of elastic couplings between the a-th and b-th subsystems, qa is the vector of displacements of the a-th subsystem, and pa is the vector of loads of the a-th subsystem (a, b – symbols of the subsystems, a  {B, F, C, S}, b  {B, F, C, S}).

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3

Aa)

Ab)

c) 1

Fmi+1

I

2

Fmk

I

c

c

I y1 I y2

... Icyi

cyi+1 ... cyk-1 cyk

I

I

Fm1

3

cz1 cz2 ... czj I

I

Fmk-1

I

I

I

Fm2

czj+1 ... czl-1 czl

I

I

Fmi

4

I

I

Fig. 1. Multi-bolted system: (a) scheme; (b) description of spring properties; (c) FEM-model with rigid bolt models (1 – subsystem B, the bolts; 2 – subsystem F, the flexible flange element; 3 – subsystem C, the conventional contact layer; 4 – subsystem S, the flexible support).

The preloading process of the system consists of k steps, in pursuance of the number of bolts in the connection. During the first bolt preloading, the system is composed of the flange element and the support, and the nonlinear contact layer between them. In this first step, the system is assembled by only one bolt and loaded by the force Fm1 which is the preload of the bolt No. 1 (Fig. 2). cy2

Fm2 cy1

Fm1

cy1 Fmi

Fmi+1 cy1

...

cyi

cyi+1

cy2 ...

cy1

cyi

cyi+1

cy2 Fmk-1

cy1

Fmk

cy2 cyk-1

cyi

Fig. 2. Sequential preloading of the multi-bolted system.

In the next steps of the system preloading (for i = 2, …, k), the next rigid body bolt model is taken into consideration. Therefore, in the Eq. (2) the stiffness matrix of the bolts subsystem KBB has been complementing with

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the elements, which are assembled in the current step of computations. However, in the current step of the system preloading, the stiffness matrix KBB is a constant part of the stiffness matrix K, and the stiffness matrices KCC and KCF are a changing part of the stiffness matrix K. 3. Calculations of the exemplary multi-bolted system on the assembly state Exemplary calculations are performed for a selected asymmetrical multi-bolted system shown in Fig. 3. 5(5)

4(2)

I

I

I

3(6)

I

6(3)

2(4)

I

I

1(1)

I

7(7)

Fig. 3. FEM-based model of the multi-bolted system with the adopted numbering of the bolts.

Characteristics of nonlinear springs in the contact layer are described as the following power function [1]: R j  A j  (3.428  u1.j657 )

(3)

where Rj is the force in the centre of the j-th elementary contact area (for j = 1, 2, 3, …, l), Aj is the j-th elementary contact area, and uj is the deformation of the j-th spring element. For assembling the system, the M10 bolts are used. The preload of the bolts Fmi is equal to 20 kN. The preloading sequence taken here on, is parenthesized in Fig. 3. Force in the bolt, kN

20.5 20.4 20.3 20.2 20.1 20 19.9 19.8

1

2

3

4

5

Number of the bolt

6

7

Fig. 4. Preload values at the end of the preloading process.

Scatter of the final bolt forces at the end of the preloading process is presented in Fig. 4. Based on the obtained results it can be stated that the preload values of the bolts during the preloading process and at the end of it are highly variable and irregular. The assessment of final values of the preload values of the bolts can be executed on the basis of the A index:

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 A

F fin ,i  Fmi Fmi

100 %

5

(4)

where Ffin,i is the value of the final preload of the i-th bolt at the end of the assembly state, and Fmi is the initial value of the preload of the i-th bolt at the beginning of the assembly state. According to the introduced model of the multi-bolted system preload values of bolts may vary in the range of -0.5 to 2.5 % in relation to their initial values.

4. Conclusions The presented model of the multi-bolted system can be successfully used in analysis of preload variations in the case of any joint in which a flexible flange element is connected with a flexible support. The model can also allow to analyse how the tightening sequence affects the preload values in bolts before the preloaded joint is loaded by an external force.

References [1] R. Grzejda, Modelling nonlinear preloaded multi-bolted systems on the operational state, Engng. Trans. 64(4) (2016) 525–531. [2] M. Abid, A. Khan, D.H. Nash, M. Hussain, H.A. Wajid, Simulation of optimized bolt tightening strategies for gasketed flanged pipe joints, Procedia Engineer. 130 (2015) 204–213. [3] R. Grzejda, Modelling nonlinear multi-bolted connections: A case of the assembly condition, Proc. of the 15th International Scientific Conference “Engineering for Rural Development 2016“, Latvia University of Agriculture, Jelgava, 2016, pp. 329–335. [4] L.Z. Zhang, Y. Liu, J.C. Sun, K. Ma, R.L. Cai, K.S. Guan, Research on the assembly pattern of MMC bolted flange joint, Procedia Engineer. 130 (2015) 193–203. [5] Y. Goritskiy, K.V. Gavrilov, Y.V. Rozhdestvenskii, A.A. Doikin, A numerical model of mechanical interaction between rough surfaces of tribosystem of the high forced diesel engine, Procedia Engineer. 129 (2015) 518–525. [6] H. Augusto, J.M. Castro, C. Rebelo, L.S. Da Silva, Characterization of the cyclic behavior of the web components in end-plate beam-tocolumn joints, Procedia Engineer. 160 (2016) 101–108. [7] X. Chen, G. Shi, Finite element analysis and moment resistance of ultra-large capacity end-plate joints, J. Constr. Steel Res. 126 (2016) 153– 162. [8] H.J. Duan, J.C. Zhao, Z.S. Song, Effects of initial imperfection of bolted end-plate connections in the reliability of steel portal frames, Procedia Engineer. 14 (2011) 2164–2171. [9] M. Alibrahemy, S. Durif, P. Bressolette, A. Bouchaïr, Finite element analysis of cover plate joint under ultimate loading, Procedia Engineer. 156 (2016) 16–23. [10] F. Ascione, A preliminary numerical and experimental investigation on the shear stress distribution on multi-row bolted FRP joints, Mech. Res. Commun. 37(2) (2010) 164–168. [11] A. Jaya, U.H. Tiong, G. Clark, The interaction between corrosion management and structural integrity of aging aircraft, Fatigue Fract. Eng. M. 35(1) (2012) 64–73. [12] R.K. Mourya, A. Banerjee, B.K. Sreedhar, Effect of creep on the failure probability of bolted flange joints, Eng. Fail. Anal. 50 (2015) 71–87. [13] J. Kortiš, L. Daniel, M. Handrik, The numerical analysis of the joint of the steel beam to the timber girder, Procedia Engineer. 91 (2014) 160–164. [14] Z. Li, K. Soga, F. Wang, P. Wright, K. Tsuno, Behaviour of cast-iron tunnel segmental joint from the 3D FE analyses and development of a new bolt-spring model, Tunn. Undergr. Sp. Tech. 41 (2014) 176–192. [15] Y. Luan, Z.-Q. Guan, G.-D. Cheng, S. Liu, A simplified nonlinear dynamic model for the analysis of pipe structures with bolted flange joints, J. Sound Vib. 331, 2 (2012) 325–344. [16] J. Aguirrebeitia, M. Abasolo, R. Avilés, I.F. De Bustos, General static load-carrying capacity for the design and selection of four contact point slewing bearings, Finite element calculations and theoretical model validation, Finite Elem. Anal. Des. 55 (2012) 23–30. [17] P. Palenica, B. Powałka, R. Grzejda, Assessment of modal parameters of a building structure model, in: J. Awrejcewicz (Ed.), Dynamical Systems, Modelling, Springer Proceedings in Mathematics & Statistics. 181 (2016) 319–325. [18] B. Błachowski, W. Gutkowski, Effect of damaged circular flange-bolted connections on behaviour of tall towers, modelled by multilevel substructuring, Eng. Struct. 111 (2016) 93–103. [19] R. Grzejda, Modelling nonlinear multi-bolted connections, A case of the operational condition, Proc. of the 15th International Scientific Conference “Engineering for Rural Development 2016“, Latvia University of Agriculture, Jelgava, 2016, pp. 336–341. [20] R. Grzejda, New method of modelling nonlinear multi-bolted systems, Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues, Proc. of the 3rd Polish Congress of Mechanics (PCM) and 21st International Conference on Computer Methods in Mechanics (CMM), CRC Press/Balkema, Leiden, 2016, pp. 213–216.