A non-linear finite element model with contact elements was developed to evaluate the contact state of a bolt - nut - washer - compressed sheet joint system. 
Applying the proper material law the non-linear behaviour of the members of the joint was studied in term of the clamping force. Based on the FE results the load distribution among the threads in contact and the real preload diagram of the system were evaluated. 
To produce the required clamping force at medium strength bolts it is advisable to use heat-treated washers instead of lower strength ones

JOANOVICS, L.

VÁRADI, K.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODlCA POLYTECHNlCA SER. MECH. ENG. VOL. 39, NO . .2, PP. 151-162 (1995) NON LINEAR FINITE ELEMENT ANALYSIS OF THE CONTACT, STRAIN AND STRESS STATES OF A BOLT - NUT - WASHER - COMPRESSED SHEET JOINT SYSTEM Laszl6 JOANOVICS and Karoly VARADI Institute of Machine Design Technical University of Budapest H-1521 Budapest, Hungary E-mail: varadi@inflab.bme.hu Phone: (361) 463-1111 Received: Oct. 8, 1994 Abstract A non-linear finite element model with contact elements was developed to evaluate the contact state of a bolt - nut - washer - compressed sheet joint system. Applying the proper material law the non-linear behaviour of the members of the joint was studied in term of the clamping force. Based on the FE results the load dis-tribution among the threads in contact and the real preload diagram of the system were evaluated. To produce the required clamping force at medium strength bolts it is advisable to use heat-treated washers instead of lower strength ones. Keywords: bolted joint, clamping force, finite element analysis, contact problems. Introduction Preloaded bolted joints are widely used to provide safe connection between different parts of a product. There are more and more high strength bolted joint applications with the requirement of the higher preload. To have an economic bolted joint the average stress at the tensile stress area As is near to the yield strength [1], [2]. Related to the modern bolted joint design there are some questions to be answered. How can we describe the behavior of the total bolt - nut -washer - compressed sheet system in the range of the higher preload? What element of the system is the critical one? Is the first thread connection always the most loaded one? What is the real load distribution among threads in contact and the real preload diagram if the members of the joint have different strength properties? What is the role of the washer in the range of the higher preload? In the traditional bolted joint design linear spring models are used [1], [3]. The members of the joint are modelled by tensioned or compressed 152 L. JOANOVICS and K. VARADI rods having equivalent diameters and assuming linear elastic material law. This approach provides reasonable results only in the low load range. To evaluate the load distribution in the elastic-plastic range [4] an optical measurement procedure is introduced. The current finite element analysis follows different non linear mate-riallaws of the members and the real geometry is studied using frictionless contact elements. (This approach produces an error proportional to the magnitude of the coefficient offriction.) [5]. The analysis covers the strain and stress analysis in a bolted joint in case of different materials, studies the role of the lower strength and heat-treated washers, furthermore eval-uates the real preload diagram of the bolted joint system. 2. Describing the Analysed Bolted Joint The sizes of bolts at the analysed models (Fig. 1) are M16x75 and M16x80. The fit of the threads is 6H/6g. The total thickness of the compressed sheets is 60 mm. The surfaces of the two sheets are parallel ideal planes. Based on the structural model the following cases are examined: Ml. bolt and nut made of material 5.8 with lower strength washer (LSW) M2. bolt and nut made of material 5.8 with two heat-treated washers (HTW) M3. bolt and nut made of material 8.8 with lower strength washer (LSW) M4. bolt and nut made of material 8.8 with two heat-treated washers (HTW) The upper part of Fig. 1 relates to cases M2. and M4., while below the symmetric line relates to cases Ml. and M3. The materials of the individual members are as follows: sheets Fe 275 (ReH/Rm = 235/380 MPa) bolts 5.8 (ReL/ Rm = 400/520 MPa) 8.8 (Rp02/ Rm = 640/800 MPa) nuts 5. 8. washers Fe 235 (ReH/Rm = 225/340 MPa) C 45 (Rp02/ Rm = 420/800) where ReL lower yield strength, Rp02 proportional limit, Rm ultimate strength. (lower strength) (medium strength) (lower strength) (medium strength) (lower strength) (heat-treated) The preload force was increased up to and beyond FY (Fig. 13) dur-ing the non-linear elastic-plastic analysis. At this load level, assuming ho-mogeneous tensile stresses, we have reached the yield strength at the ten-XON LINEAR FINITE ELEMENT ANALYSIS 153 l2 = 9 ·-+------'--6-0-----1\ A1 = 201.; mm2 Fig. 1. sile stress area (As). In the technical literature there are different allowable working load levels (based on homogeneous stress calculation) depending on the operating conditions. The load level of 0.55 Rm and 0.62 Rm (see Fig. 13) corresponds to the average and maximum preload of a general purpose bolted joint [1]. Considering the operating conditions there are circumstances when the working load in the bolted joint may reach Fproof. (F proof is the highest tensile force they could withstand without taking any permanent deformation [1]). 3. Stiffness Calculation of the Bolted Joint Based on Linear Spring Models The following equations represent the traditional analysis of the bolt - nut - washer - compressed sheet system [1], [3]. The total stiffness KT of bolt, nut and washer: Cases Ml and M3 Cases M2 and M4 _1 ___ 1_ _1_ __1_ KTl - KBl + KN + Kw!' 1 _ 1 1 2 KT2 - KB2 + KN + KW2' 154 L. JOANOVICS and K. VARADI The bolt stiffness KB [3], following the notation of Fig. 1: 1 1 [ d 11 12 d ] KB = E 0.4 A1 + Al + Am + 0.4 Am . The stiffness of the nut KN and washer Kw is directly calculated from the geometry. The stiffness of the compressed sheets KJ was calculated according to MEYER - STRELOW [3], where the equivalent cross-section is Ac = 630.02 mm2 • The calculated stiffnesses: KBl = 5.2828.105 N/mm, KB2 = 4.9718 .105 N/mm, f{N = 4.3982.106 N/mm, KWl = 3.3215 .107 N/mm, f{W2 = 2.4912 .107 N/mm, f{J = 2.2051.106 N/mm, KT! = 4.6503,105 N/mm, KT2 = 4.3122.105 N/mm. The preload diagrams based on these stiffnesses are shown with thin lines in Fig. 13. 4. The Finite Element Model of the Bolted Joint To model the bolt nut washer - sheet system 4 node axisymmetric elements are used. Frictionless contact elements are located between the threads in contact, on both sides of the washers and between the bolt's head and the sheet in the following numbers: threads: 6 x 5 = 30 elements, washer: 11 + 17 = 28 elements, bolt's head: 11 elements. Fig. 2 shows the axisymmetric finite element model for cases M2. and M4., while the enlarged FE mesh in the vicinity of the bolt, nut and washer is shown in Fig. 3. The clamping force is introduced as distributed load over the cross-section in the middle of the shank of bolt (the bolt was cut into two parts). The clamping force 'flows' through the nut, washer, sheets and bolt's head, producing a closed force flow. The material laws are linear static and strain hardening ones accord-ing to the given yield strength of each member. The tangent modulus of each material in the strain hardening range is E' = 500 MPa. The boundary conditions are activated at the common surface of the compressed sheets (Fig. 1) allowing only radial displacement. To solve this problem the non-linear module of COSMOS/M EX-PLORER and the displacement control algorithm were used. This algo-rithm could evaluate the fairly high plastic strains in the lower strength washers. MP CLR 1 _ 2 _ . -5 _ s II1II J NON LINEAR FINITE ELEMENT ANALYSIS Fig. 2. 5. The Stresses and Strains in the Bolted Joint 5.1 Bolted Joint with Lower Strength Bolt (5.8) and Lower Strength or H eat- Treated Washers 155 The stresses and strains of a lower strength bolt and nut are shown in Figs. 4, 5, 6 and 7 at 0.62 Rm load level. By the help of the deformed shapes the local sliding between the threads and the smaller radial sliding of the washer can be followed in Fig. 4. The von-Mises equivalent stress distribution shown in Fig. 4 explains the 'force flow' producing overloading between the first threads near to the washer, furthermore local overloading in the bolt next to the washer. Altogether these results show small plastic zones in the bolt and the nut. The stress levels are lower in the washer and compressed sheets due to the lower yield strength. In Fig. 5 the high stresses in the vicinity of the bolt's head are shown. The plastic behaviour of the analysed bolted joint system is shown in Figs. 6 and 7. In the black areas the total equivalent strains (definition 156 i-. JOANOVICS and K. VARADI MP CL" I 1 -2 3 -• 5 -6 I I I I I I I I I i I i I I I I I i ! I I I I I I I I I I I I I I , I I I I I I I I I I i I I \ J~ ~\ A AI \ , \ I I I 1\\ ,\ I I \t , /7 I \ \ I \ \ I \ \ / I \\ I I \ \ / I I " \ I I t"-fl I I I I I I I I I I I I I I r I I TT, i I I I I I I I I i ! I I I " I I I , I I ' Fig. 3. in [6]) are greater or equal to 0.002 (0.2 %). This equivalent strain level represents the border of plastic zones of the bolt and the nut, because the yield strength of them is 400 MPa. At this load level there are smaller plastic zones in the vicinity of the threads and in the nut next to the washer. The plastic zone in the washer is slightly greater than the black area in Fig. 6 due to the lower yield strength (235 MPa) of the washer. The heat-treated washer in Fig. "/ remains totally in elastic range at 0.62 Rm load level. 5.2 Bolted Joint with Medium Strength Bolt (8.8) and Lower Strength or Heat- Treated Washers Let us use the same load level of 0.62 Rm. In Fig. 8 the black areas represent the total equivalent strains, that is greater or equal to 0.0032. (Yield strength is 640 MPa). The extent of the plastic zones in the vicinity of the threaded zones is almost the same as in Fig. 6 because the load level is proportional to the higher ultimate (or yield) strength. In the lower strength washer at the same time there is a big plastic zone with the maximum equivalent strain of 0.029 (2.9 %). The huge NON LINEAR FINITE ELEMENT ANALYSIS 157 ~ ~ ~ 7 lsF 158 L. JOANOVICS and K. VARADI :'! ~ ! JJ ;. !. !. , ::;; ''0;;/.:,>; s -= -CJl CJl 1 lE Lt: 1 1 I I I I I" I~, I~I "j 11 NON LINEAR FINITE ELEMENT ANALYSIS 159 plastic zone basically modifies the stiffness of the washer and consequently the stiffness of the whole system. The plastic zones in the compressed sheets are smaller. In Fig. 10 near to the bolt's head we have a bigger local plastic zone in the compressed sheet with the maximum equivalent strain of 0.006 (0.6 %). This plastic zone modifies the stiffness of the sheets and consequently the stiffness of the whole system. The role of the two heat-treated washers is shown in Figs. 9 and 11. N one of the washers enters into the plastic state and the load is transmitted via a larger surface. 6. Load Distribution among the Threads in Contact Let us analyse thread by thread the force system transferred through the contact elements between the threads in contact. In Fig. 12, following the order of the finite element mesh, the loa,: distributions for cases M1 and M3 are shown for both materials. The numbers represent the load portion related to the ultimate strength. The first series of curves (from left to the right) characterise the behaviour of the threads in the elastic range, while the next curves show the effect of the plastic zones near to the most loaded threads. According to the elastic load distribution (first curve in Fig. 12) the load on the first thread is about the double of the load of the last thread contrary to the difference of 3 to 4 given in a textbook published earlier [7]. To check this model for one elastic calculation we fixed the nut surface (next to the washer) in axial direction, while the radial displacement was allowed. The results drawn by dotted lines show a good agreement to the earlier results. Let us study the load distribution in higher load range. Near to the load level equivalent to 60 % of the ultimate strength of the bolt the most loaded thread cannot transfer higher load, due to the increasing plastic zones, while the following threads can do it. At about 70 % of the ultimate strength the second thread cannot transfer higher load any more. The third thread has the same performance above the load equivalent to 80 % of the ultimate strength. 7. The Preload Diagram of the Bolted Joint Fig. 13 shows the preload diagram for the studied four cases. Thin lines denote the preload diagram at level of 0.55 Rm obtained by linear spring model [1]. 160 L. JOANOVICS and K. VARADI ~~--~~~-<~u-~----~ F CDI.=======~~~~~~ I M16x2 6H/6g j s.S-s·1 G)~======~-b--~~~ CD CD CD CD CD CD CDr---------~~----------~~------~ I H16x2 6H/6gj B.S-8. I 0~======~~~~~~~------~ o 5000 10000 Fig. 12. 15000 FIN] Considering material 5.8 both the lower strength (LSW) and the heat-treated washers (HTW) have almost the same stiffness behaviour. There are bigger differences considering material 8.8. In case M3. (8.8+LSW) at about preload level of 0.55 Rm there are huge plastic zones in the lower strength washer and the compressed sheet (near to the bolt's head). In this way the better material of the bo~t. and nut cannot be utilized, the re-NON LINEAR FINITE ELEMENT ANALYSIS 161 I~ E .§ <i "" I=:, o· E ' '" I J;l1~ i .; I L-..J I'" i I :r: i I I I • I 1-I l"J I" :;::. I I I I I I I I 1 T~ I 10' I I i I , ~ I -' ,0 0.; ~ ~I ..... u. E .~ I I E "'" I I "]" I I VI i I ! ~ ! ~I ~i r ~ l ! 0 ","'l I x I i "" i i '-I .rl I E si ~~! i Q: "'j I ... . < <I !~. x x ~j~! I I 11 n! E u.: Ll.; ~1~ i i I , I i i ::; i N ' I i ~~ I I L:J ' I I I , ! I I ! ! I i -t", ,0 l...i "', I ''''' ::::! ,!d I It; ]! ~e : I , CL u.j I iU: r " I I ~ '0 ' - '0 I~ '8 to 8 =. g 0 u. ~ u.. R '" 162 L. JOANOVICS and K. VARADI quired preload cannot be achieved. Two heat-treated washers can solve this problem. In case M4. (8.8+HTW) the preload diagram is almost linear up to the range of the yield strength so this preload level may be achieved. 8. Conclusions The non-linear axisymmetric model with contact elements is suitable to analyse the bolted joint having parts with different strength prop-erties, to find the plastic zones, to obtain the load distributions among threads in contact, to determine the real preload diagram of the bolted joint system. To be able to increase the preload of a bolted joint system having medium strength bolts and nuts it is advisable to use heat-treated washers. References 1. BICKFORD, J. W.: An Introduction to the Design and Behavior of Bolted Joints, Second Edition, Marcel Decker, N. Y. -Basel, 1990. 2. VRAUKO, L.: Gepipari kotoelemek alkalmazasa a tervezesben, gyartasban es szereles-kor, Szabvanykiad6 Budapest, 1985. 3. MEYER, G. - STRELOW, D.: Simple Diagrams Aid in Analysing Forces in Bolten Joint, Assembly Eng., January, 1972. 4. STOCKMANN, M. - ENGELMANN, G. - NAUMANN, J.: Analysis of Elastic-Plastic De-formations in Screw Threads by Optical Measuring Methods, VDI Berichte Nr. 940, 1992. 5. V ARADI, K.: TerheIesatad6 gepelemek erintkezesi es fesziiltsegi allapota, micro CAD-SYSTEM'93, Miskolc, 1993, marcius 2-6. 6. COSMOS/M, Version 1.70, User Guide, Structural Research and Analysis Corporation, 1993. 7. VOROS, I.: Gepelemek I., Tankonyvkiad6, Budapest, 1970. 163 Gyula Strommer Es hatte ein Festtag fur Professor Gyula Strommer werden sollen, und ist doch ein Trauer- und Gedenktag geworden. Herr Prof. Strommer ist am 28. August von uns gegangen, nachdem ein unerbittliches Schicksal ihn schon monatelang an das Krankenbett gefesselt hatte. Wer das Gliick hatte, Herrn Strommer naher gekannt zu haben, hat viel von ihm lernen konnen. Seine wissenschaftlichen Leistungen, vor allem auf dem Gebiet der Ax-iomatik, haben ihn weit uber die ungarischen Grenzen hinaus bekannt gemacht. Wer erinnert si ch nicht an seine brillianten Vortrage, bei denen er in fehlerlosem Deutsch seine Ideen vortrug, und dabei in ruhigen, wohlgemessenen Schritten vor der Tafel oder Lein-wand auf- und abwanderte. Aber auch seine Lehrbucher haben ihm verdienten Ruhm eingebracht. Und Tausende der heute aktiven ungarischen Ingenieure sind durch ihn in die Gedankenwelt der Darstellenden Geometrie eingefuhrt worden. Aber auch seine menschliche Seite war vorbildhaft. Er war voU frellndlicher Vornem-,heit und Korrektheit. Er wllBte sehr wohl, ehrliches wissenschaftliches Bemuhen von Scharlatanerie zu unterscheiden. Es fehlte ihm allch nicht ein Hang Zll leicht ironischer Selbstkritik. Aber er war auch oft voll Lebensfreude, und man konnte viellachen mit ihm, obwohl er in seinem Leben auch mit so manchen Harten fertig werden muBte. Der allzll fruhe Tod von Prof. Gyula Strommer schmerzt. Ein Grandsegnieur der Geometrie hat uns fur immer verlassen. Er.wird uns fehlen. Hellmuth Stachel im Namen der Tagllngsleitllng der Gedenktagung Konstruktive Geometrie Balatonfoldvar 11-15 September 1995. 

NON LINEAR FINITE ELEMENT ANALYSIS OF THE CONTACT, STRAIN AND STRESS STATES OF A BOLT - NUT - WASHER - COMPRESSED SHEET JOINT SYSTEM

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NON LINEAR FINITE ELEMENT ANALYSIS OF THE CONTACT, STRAIN AND STRESS STATES OF A BOLT - NUT - WASHER - COMPRESSED SHEET JOINT SYSTEM

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Periodica Polytechnica (Budapest University of Technology and Economics)