Out-of-Plane Loading Effects on Slip-Critical Screw Joints. Master s thesis in Applied Mechanics MARCUS KARLSSON

Transcription

1 Out-f-Plane Lading Effects n Slip-Critical Screw Jints Master s thesis in Applied Mechanics MARCUS KARLSSON Department f Applied Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Götebrg, Sweden 2016

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3 MASTER S THESIS IN APPLIED MECHANICS Out-f-Plane Lading Effects n Slip-Critical Screw Jints MARCUS KARLSSON Department f Applied Mechanics Divisin f Slid Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Götebrg, Sweden 2016

4 Out-f-Plane Lading Effects n Slip-Critical Screw Jints MARCUS KARLSSON MARCUS KARLSSON, 2016 Master s thesis 2016:38 ISSN Department f Applied Mechanics Divisin f Slid Mechanics Chalmers University f Technlgy SE Götebrg Sweden Telephne: +46 (0) Cver: Blted Jint Mdel Chalmers Reprservice Götebrg, Sweden 2016

5 Out-f-Plane Lading Effects n Slip-Critical Screw Jints Master s thesis in Applied Mechanics MARCUS KARLSSON Department f Applied Mechanics Divisin f Slid Mechanics Chalmers University f Technlgy Abstract This thesis aims t prvide a thery fr the influence f ut-f-plane lads n the slip cnditin f screw jints. This thery is develped using rigid bdy thery and after the frces n the plates have been fund, the cncept f elastic clamping areas is intrduced and a frmula fr material stiffness develped. These expressins fr the slip resistances will be used t find the critical lad fr a ttal jint slip. The thery will then be cmpared t a Finite Element (FE) mdel that is parametrised t vary the mment arm with respect t the length ut f the plane. The thery shws a very lw influence f the ut-f-plane lad n the ttal slip resistance f the jint while the FE simulatins shw a significant increase. Using very stiff plates in the FE simulatins hwever, yields the same results as the thery, leading the authr t believe that the increased slip resistance is a gemetrical effect. The blt frces are increased significantly in the blts lcated in the area experiencing separative frces. This effect must be taken int cnsideratin when designing jints with lad applied ut f the plane t avid in-field failure due t blts ging past yield lad. Keywrds: Blted Jints, Screw Jints, Slip-Critical Applicatins, Blt/Screw Jint Eccentric Lading i

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7 Preface This is a Master s Thesis fr an M.Sc. degree in Applied Mechanics at Chalmers University f Technlgy. The wrk was carried ut during the spring f 2016 at the Analytics department f Alten Sweden AB fr the custmer Ålö. M.Sc. Mikael Almquist was the supervisr at Alten and Ph.D. Carl Sandström was the supervisr at Chalmers. Magnus Ekh, Prfessr at the divisin f Material and Cmputatinal Mechanics at Chalmers, was the examiner. Acknwledgements Thanks t all the clleagues at Alten fr the cmpany and supprt. Special thanks t Carl Sandström, my supervisr, fr his supprt and cheerful attitude. Thanks t Mikael Almquist fr giving me the pprtunity t wrk with this prject and thanks t Daniel Rickert fr great discussins arund mechanics and help with varius sftware prblems. I wuld like t thank Jhan Grankvist, wh has been my cntact at Ålö, fr his ideas and weekly fllwups and fr the chance t visit in Umeå. iii

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9 Nmenclature α Angle f F IP t x-axis [rad] d b Elngatin f blt [m] d c Cmpressin f clamped parts [m] F b Change in blt frce [N] F c Change in clamping frce [N] δ Deflectin [m] ɛ Kinematic strain γ Kinematic strain κ Kinematic angle µ Frictin cefficient [ ] σ blt,i Stress in blt crss sectin [N/m 2 ] σ fr Frictinal stress [N/m 2 ] σ mean,i Mean stress in blt crss sectin frm parallel axis therem [N/m 2 ] σ n Nrmal stress [N/m 2 ] θ Angle f material defrmatin frustum d i Distance frm centre f gravity fr jint [m] E Yung s mdulus [N/m 2 ] e Eccentricity f in-plane lad [m] F a F b F c F s F T F blt,i F bs F change F cs F IP F mean,i F OP F pre F P F r i External axial lad [N] Tensile frce in the blt [N] Clamping frce between the plates [N] Maximum frictin frce [N] Transverse lad [N] Axial frce in blt [N] Frictin frce in blt head plane [N] Change in blt and clamping frce [N] Frictin frce in clamped plane [N] In-plane frce [N] Frce n blt frm parallel axis therem [N] Axial frce frm external mment [N] Blt pre-lad frce [N] Transverse frces n clamped areas [N] Resulting frces frm in-plane frces [N] Blt index I par,x Area mment f inertia [m 4 ] K b K c K f Blt stiffness [N/m] Stiffness f clamped material [N/m] Flexural stiffness [N/m] v

10 k s m M IP M OP n P Frictin cefficient f faying surface Number f faying surfaces (slip planes) In-plane mment [Nm] Out-f-plane mment [Nm] Number f blts Critical lad [N] P avg Pressure [N/m 2 ] r 0 r i R s R slip s fac T i x i y y i z FE I.C. RBE Distance t centre f rtatin [m] Radial distance frm assumed centre f rtatin [m] Kulaks slip resistance [N] Slip resistance [N] Safety factr fr shear frce in clamp area Clamping frce [N] Blt x-crdinate [m] Distance frm mment applicatin [m] Blt y-crdinate [m] Crdinate frm single blt centre f gravity [m] Finite Element Instant centre f rtatin crdinate Rigid Bdy Element vi

11 Cntents Abstract Preface Acknwledgements Nmenclature Cntents i iii iii v vii 1 Intrductin Limitatins Review f Cntact Mechanics in FEM Gap Functin Culmb Frictin Mdel Separatin FE Mdelling Thery Failure Criteria Blt Mdelling Slip Cnditin Rigid Bdy Mtin Jint Frces Out-f-Plane Mment In-Plane Mment In-Plane Frce Resulting Frces Wrst Blt Mdel Ttal Jint Slip Elastic Members Clamping Frce Effect f Axial Frces n Slip Resistance Finite Element Analysis Cntact Pressure Slip Detectin Critical Lads Mdel Mdel Mdel Small Displacements Stiff Plates Observatins Blt Frces Real Mdel Simulatin Discussin Future Wrk References 28 vii

12 Appendices 29 A Jint Designer Prgram 30 A.1 Interactive Script A.2 Parameter Studies A.2.1 Wrst Blt Mdel Predictins A.2.2 Varying Clamped Material Stiffness B Mtivatin f Rigid Bdy Equatins 32 B.1 Area Mment f Inertia B.2 Out-f-plane Mment Frces B.3 In-plane Mment Frces C Autmatin f Pst-Prcessing 35 D R Cde 35 viii

13 1 Intrductin Jining parts tgether using screws r blts is an easy and cmmn way t assemble a structure. When designing these jints it is imprtant t knw abut their limit lad befre failure, s that unexpected failure in the field des nt ccur. The many failure mdes f such jints make it an inherently difficult jint t analyse. One f the challenges is t design these jints against slip, a seemingly menial task, but as the lad cases becme mre cmplex, the cmplexity f the jint behaviur increases rapidly. One such lad case is when the lad is eccentrically placed ut f the plane f the jint, further cmplexity is intrduced if the lad is eccentrically placed in the plane. This thesis aims t investigate the effects n the slip cnditin when placing these lads eccentrically ut f the plane. In Figure 1.1 a mdel f a tractr frnt lader subframe is displayed, shwing hw the lad is applied ut f the plane f the screw jints. Lad Lad Lad (a) Ismetric view f the mdel (b) Out-f-plane eccentricity visible Figure 1.1: Mdel f tractr subframe split in symmetry plane (c) In-plane view f the jint FE simulatin is an excellent tl in this type f analysis since regular hand calculatins are nt easily applicable n such a cmplex prblem as sliding n large elastic areas. The gal f this prject is t develp a mdel that can incrprate an ut-f-plane mment int a mdel t predict slipping lads. This mdel shuld be implementable in such a way that a designer can input jint data and get apprximate critical lad and slip cnditins in the blts. Initially a thery using rigid bdies will be develped. This mdel will then be used t find the slip cnditins f the jint and clamped areas. A simplified FE mdel f a jint will be made t cmpare the results and crrelate with the develped thery. The discretised mdel will be slved using MSC s NASTRAN slver SOL400 [1]. It is assumed that finite defrmatins exist due t the sliding cntacts and the high lads t vercme the clamping frictin f the blts. A mdel f a real jint will be cmpared t the thery t see if it can be applied t real wrld applicatins. 1.1 Limitatins The prject will have sme limitatins and assumptins regarding mechanical behaviur will be made. Linear elastic material will be used in the FE analysis and thery Static cnditins Fracture and yielding will nt be cnsidered 1

14 1.2 Review f Cntact Mechanics in FEM The theretical mdel will be cmpared t FE simulatins, where large variatins can ccur depending n the settings used. It is therefre imprtant t define what settings that are used thrughut the simulatins Gap Functin When intrducing cntact mechanics, a cntact functin t determine when ndes are in cntact is needed. The nde-t-segment cntact mde is chsen, meaning that the ndes will be checked against the surface f the ppsing elements. The ptin t use segment-t-segment cntact checking is pssible but this functin is still under develpment in SOL400 [2] and therefre the nde-t-segment cntact is used Culmb Frictin Mdel The Culmb frictin mdel has been chsen in the FE simulatins. The MSC.Nastran dcumentatin has sme ntes n the cnstitutive equatins fr Culmb frictin [3]: σ fr µσ n (1.1) where σ fr is the frictinal stress, σ n is the nrmal stress (σ n 0), µ is the frictin cefficient between the surfaces. In the Culmb mdel, a pint in cntact will be either sliding r sticking Separatin Due t hw ndal frces behave in 2nd rder elements, the integrated ndal stress will be used as the separatin cnditin [2] tgether with a tlerance. Fr the small testing mdels used in this thesis, the separatin tlerance is set t 1% f the maximum cntact stress in the mdel t achieve high accuracy. Since the separatin f ne nde can cause anther t cme int cntact again, it is required t set a maximum number f separatins fr each lad step, ften referred t as chatter. After this maximum has been reached, the individual nde will n lnger be cnsidered fr separatin. There is als a pssibility t change the ttal number f allwed separatins fr all ndes in the mdel fr a single step t further reduce chatter frm nde separatins, limiting iteratins based n the glbal number f nde separatins FE Mdelling T set up an FE simulatin frm a given gemetry sme preparatins have t be made. First the gemetry has t be cleaned, all surfaces defined and all vlumes clsed, and simplified t avid verly cmplex behaviur that might nt be the target f the analysis. Figure 1.2 displays a simplified flwchart f the steps needed t cmplete an analysis, starting frm a supplied gemetry and ending with the analysis f the simulated lad case. 2

15 Pre-prcessing Imprt gemetry Clean gemetry Mesh gemetry Check mesh quality Define cntact bdies Apply bundary cnditins Prcessing Check input mdel Run slver Mnitr cnvergence Retrieve utput file Pst-prcessing Lad input file Lad utput file Lad results t gemetry Check mdel defrmatins and stresses Extract relevant data Plt and interpret data Figure 1.2: Wrkflw fr the FE simulatins 3

16 2 Thery The aim f this thesis is t find a mathematical mdel f hw the ut-f-plane lading affects the slipping cnditin f a blted jint based n the rigid bdy thery. In this chapter the cncepts f slip and jint failure will be presented as well as sme f the mdelling techniques fr blted jints in FE analysis. 2.1 Failure Criteria The jint will be cnsidered failed when there is slip in the whle jint, meaning that the whle blt area is slipping. This will ultimately initiate cntact between the blts and the blt hle, creating a shear frce n the blt that the blt is nrmally nt designed t withstand. 2.2 Blt Mdelling Mdelling blts in FE analyses has t be very carefully perfrmed as the intrduced bundary cnditins can lead t behaviur that is far frm accurate. Sme ways t mdel blts are presented in Table 2.1 Type Prs Cns Beam Element Blt Cmputatinally cheap Des nt capture separatin Captures head and nut behaviur Slid Blt Captures separatin in cntact Cmputatinally expensive Captures thread stresses Threaded Blt If threads are cmpletely mdelled with rise it is Extremely cmputatinally expensive pssible t test trque t pre-tensin Table 2.1: Different types f blt mdelling techniques Sme illustratins f the different mdelling techniques available are displayed in Figure 2.1. The cmputatinally cheapest way t mdel blts are nly gd fr very simple analyses where the actual slip cnditin is nt f great imprtance. Since the blt head is nt there t apply anther slip plane the, slip cnditin is greatly changed. Instead the blt resists slipping mtin with its full flexural stiffness. Meanwhile the threaded blt technique is mainly gd when thread specific behaviur needs t be analysed. The slid blt mdelling captures the stresses in the blt bdy and the frictin between the blt head and the clamped part. 4

17 (a) Rigid bdy elements n the inside f the hles cnnected with beam elements (b) Multi-pint cnstraint intrduced after blt is split (d) Blt split and ndes tied with multi-pint cnstraint (e) Blt pre-stressed (c) Threaded blt (f) Blt in lading just befre slip (g) Blt pst slip Figure 2.1: Stresses and displacements fr different blt mdels in varius situatins, defrmatin scale factr f 100 During this thesis the slid blt mdelling technique will be used as this is a gd apprach fr blted jint analysis where the general blt and jint behaviur is the target f the simulatin. 5

18 2.3 Slip Cnditin The slip cnditin fr any pint in a cntact surface can be simplified t the clamping pressure multiplied by the frictin cefficient. Accrding t the Culmb frictin mdel, this is the maximum shear stress the pint n the surface can resist befre slip ccurs. In Figure 2.2 the slip cnditin f a transversely laded strip is shwn. F cs is the maximum frictin frce in the clamped plane and F bs is the maximum frictin frce in the blt head plane. F F bs F bs F b F c F b F b F cs F cs F c Figure 2.2: Blt area simplified as a strip with tw slip planes Frm this figure the maximum frictin frce F s and maximum allwed transverse lad F t fulfil static cnditins fr the strip is: F F s µf c + µf b (2.1) If the strip is laded in the axial directin f the blt, a difference in F b and F c can be shwn using frce equilibrium, this will be treated further ahead in Sectin 2.4. It is als assumed that the blt can carry the full slipping lad befre deflectin. 6

19 2.4 Rigid Bdy Mtin Assuming rigid bdy thery, the effect f the ut-f-plane lad n the clamped areas can be derived by splitting it int tw mments and ne transverse frce. First the ut-f-plane frce is derived by assuming rigid bdy mtin arund an axis thrugh the centre f the plate. The superpsitin principle is then implemented n the tw different lad cases in the plane, ne with the mment in the plane f the plate and ne with the transverse frces, t btain the final shearing frce n the clamped area cming frm the plate. The plate is cnsidered rigid and the blts are cnsidered as elastic springs Jint Frces Due t the applied external lad, reactin frces in the clamping areas must be accunted fr. Using rigid bdy thery, an expressin fr these frces depending n the applied lad can be derived Out-f-Plane Mment The ut-f-plane mment will influence the clamping frce f the separate blt areas. T find this influence, an assumptin that the mment will be distributed as a frce n all blts depending n the area f inertia is made. In Figure 2.3 the expected reactin frces can be seen. F OP,i M OP z y Figure 2.3: Out-f-plane mment effect n the blts Say that i is the blt index and n is the number f blts. The area mment f inertia arund the centre f mass, I par,x, using the parallel axis therem [4] (see Appendix B.1), is then cmputed as: I par,x = n A i yi 2 (2.2) i=1 where A i is the blt crss sectin area and y is the y-crdinate f the blt centre. The average pressure P avg,i in the blts can then be calculated as (c.f. Appendix B.1): P avg,i = M OP y i I par,x (2.3) where M OP is the mment ut f the plane induced by the applied lad. Multiplying with the blt crss sectinal area yields the blt axial frce F OP,i created frm the mment: F OP,i = P avg,i A i = A i M OP y i I par,x = A i M OP y i n A j yj 2 j=1 (2.4) An expressin fr the axial frce induced by the ut-f-plane mment, assuming that all blt crss sectins are 7

20 unifrm, can then be calculated as: F OP,i = M OP y i n yj 2 j=1 (2.5) In-Plane Mment The same prcedure used in Sectin can be used fr the in-plane mment and an expressin fr the shearing frce frm the in-plane mment achieved. Figure 2.4 describes the reactin frces in the blts frm the in-plane mment and transverse frce. Here r i is the radial distance frm the assumed centre f rtatin and α i the angle f F IP t the x-axis. P,i M IP r i F P,i F IP,i M IP r i F P,i F r,i F IP,i F P,i F IP,i M IP r i F P,i F r,i F P,i F IP,i F P,i F r,i F IP,i F T F T F T y x (a) Transverse frces (b) In-plane mment (c) Resultant frces Figure 2.4: Equilibrium fr applied frce and mment in centre f gravity fr blts α i The expressin fr the shearing frces F IP,i frm the in-plane mment, M IP, is cmputed using area mment f inertia, assuming unifrm blt crss sectin areas (see Appendix B.2 fr further infrmatin): F IP,i = M IP r i n rj 2 j=1 (2.6) where r i is dependent n the blt crdinates x i and y i as: r i = x 2 i + y2 i In-Plane Frce The transverse frce is then mved int the plane due t the mments intrduced and is assumed t be divided equally as F P,i between the blts: where F T is the applied lad mved int the plane. F P,i = F T /n (2.7) 8

21 2.4.5 Resulting Frces A vectr sum is used t determine the resulting frce F r,i acting n the clamped area in the plane. The angle α i is given by: π 2 arctan y i fr x 0, y > 0 x i π 2 + arctan y i fr x > 0, y 0 x i α i = 3π 2 arctan y i fr x < 0, y < 0 x i 3π 2 arctan y i else x i The magnitude f the in-plane frce is: 2 2 F r,i = M IP r i n cs(α i ) + F P,i n + M IP r i n sin(α i ) (2.8) rj 2 j=1 rj 2 j=1 The clamping frce F c,i can then be expressed as the prelad frce summed with the axial frce frm the ut-f-plane mment: F c,i = F pre,i M OP y i n yj 2 j=1 The slip resistance R slip,i fr any blted area can then be expressed as: (2.9) R slip,i = F c,i µ + F b,i µ (2.10) While the blt retains its pre-lad frce F pre since the plates are cnsidered rigid, n defrmatin ccurs in the blt until separatin, leading t: F c,i = F pre,i + F OP,i (2.11) 2.5 Wrst Blt Mdel F b,i = F pre,i (2.12) An expressin fr expected shearing frces and slip resistance in any blt is nw frmulated. It can then be stated that the safety factr, s fac, fr any blt is: s fac,i = R slip,i F r,i (2.13) Eq. (2.13) is expanded, arriving at a frmula fr the safety factr against slip regarding lcal shear frces, meaning that this is nt a safety factr fr applied frce but fr the shear frces in the clamping areas. s fac,i = M IP r i n rj 2 j=1 2F pre,i M OP y i µ cs(α i ) 2 + n rj 2 j=1 F P n + M IP r i n rj 2 j=1 sin(α i ) 2 (2.14) 9

22 2.6 Ttal Jint Slip Here Kulak s instant centre f rtatin thery [5] is intrduced, a mdel f when a jint experiences cmplete slip. Kulak s methd assumes that the jint will slip in all blts simultaneusly and that it is pssible t find a pint arund which the whle jint will rtate at the mment f slip. The critical lad is then reslved by finding this pint using mment equilibrium. In Figure 2.5 the lading directin and eccentricity e f the applied lad P can be seen. This will be used t find the instant centre f rtatin (I.C.), crdinate using mment equilibrium. This pint must be n a line perpendicular t the lad directin accrding t Eq. (2.16), passing thrugh the intersectin f the balance lines f the slip resistance f the clamped areas. S a balance line, shwn with dashed line in Figure 2.5, is fund by weighting the crdinates f the blts with their slip resistance and finding their balance lines in bth y-directin and x-directin. r 0 e I.C. r i φ i R slip,i y x P Figure 2.5: Diagram f frces fr mment equilibrium where r 0 is the distance frm the intersectin f the balance lines. The equilibrium equatins can thus, accrding t Kulak [5], be stated as: : : : n R s cs(φ i ) P = 0 (2.15) i=1 n R s sin(φ i ) = 0 (2.16) i=1 P (r 0 + e) n R s r i = 0 (2.17) i=1 10

23 Here, Kulak s slip resistance term R s is: R s = mk s T i (2.18) where m is the number f faying surfaces (slip planes), k s is the frictin cefficient f the surfaces and T i is the clamping frce. Kulak s thery can be extended t the case f varying slip resistance in the blts. Assuming tw slip planes between the plates and blt heads, Kulak s slip resistance will be replaced by the slip resistance in the rigid bdy mdel, where the slip resistance is: R slip,i = (F b,i + F c,i ) µ (2.19) where F b is the blt frce and F c the clamping frce. This slip resistance is valid assuming that bth planes have the same frictin cefficient and the blts carry the full slip lad befre deflectin. Intrducing this new slip resistance gives a slip resistance that is individual t each clamped area. A new balance line must therefre be defined using the new slip resistances as weights placed n the blt lcatins. These new balance lines, are defined in the x- and y-directin. Using equilibrium fr the system where the frces n the blts are chsen t be the slip resistance R slip,i, acting perpendicular t the I.C., describes the state right befre slip. Here, it is pssible t slve fr the critical lad P by iterating the pint r 0. n P 1 = R slip,i cs(φ i ) i=1 P 2 = n R slip,i r i i=1 (r 0 + e) (2.20) When P 1 P 2 tl the iterating is terminated and the critical lad is determined by the equilibrium equatins. It is imprtant t nte that as the eccentricity f the lad is appraching zer, i.e. the intersectin f the balance lines, the equatins (2.20) will be appraching the transverse slip resistance as can be seen in Figure 2.6a, meaning that n rtatin will ccur and the instant centre f rtatin will be infinitely far away. Figure 2.6b illustrates this singularity as the eccentricity appraches zer. 11

24 Critical Lad fr Varying Eccentricity f In Plane Lad Instant Center x crdinate Critical lad [N] Critical Lad Transverse plane slip resistance Distance f Instant Center frm CG In plane eccentricity f lad [m] In plane eccentricity f lad [m] (a) Critical lad with varying eccentricity (b) Distance f instant centre f rtatin frm balance line intersectin Figure 2.6: Singular and asympttic behaviur f the instant centre f rtatin methd 2.7 Elastic Members The frces frm the rigid bdy theries can be applied t the assumptins that the plates and blts are elastic nly in the clamping areas and rigid elsewhere. Using an expressin fr the material stiffness develped herein, a relatinship between the external axial frces, the blt frces and clamping frces are develped Clamping Frce A clamped area can be described as a system f springs in parallel where the material(s) being clamped is in series. T slve the equilibrium in this system the stiffness f the material is idealised as springs. It is assumed that all material experiencing defrmatin cntributes t the stiffness nly in the clamping directin. The material in the clamped areas can be assumed t defrm in the shape f a frustum, as shwn in Figure 2.7a, with a 30 angle, θ, accrding t Shigley [6]. Figure 2.7b shws a schematic representatin f the spring system, where K B is the blt stiffness, K c1 and K c2 is the material stiffness, r 0 is the hle radius, r 1 is the blt head radius and r 2 is the radius at depth x. 12

25 r 1 F b K c1 F c x K b F c F a K c2 r 2 (x) (a) Defrmatin cne (b) Spring system f clamped area Figure 2.7: Defrmatin cne and schematic f spring system fr clamping area The axial defrmatin δ f a crss sectin A is used t find the stiffness f the frustum: dδ = P dx EA (2.21) where E is the Yung s mdulus f the material and P is the applied lad. In the frustum a varying crss-sectinal area alng the axial directin is expressed as: A = π ( r 2 2 r 2 0) = [r2 = r 1 + x tan(θ)] = = π (x tan θ + r 1 + r 0 ) (x tan θ + r 1 r 0 ) (2.22) The equatin Eq. (2.21) can nw be frmed as the integral: δ = P L 1 dx (2.23) πe 0 (x tan θ + r 1 + r 0 ) (x tan θ + r 1 r 0 ) Integrating Eq. (2.23) accrding t Mathematics Handbk [7] pg.156 Eq.49 yields the equatin fr elngatin: The equatin fr stiffness is: δ = P πe2r 0 tan θ ln (L tan θ + r 1 r 0 ) (r 1 + r 0 ) (L tan θ + r 1 + r 0 ) (r 1 r 0 ) k = P δ (2.24) (2.25) The final stiffness K c fr the clamped material thus becmes: K c = πe2r 0 tan θ ln (L tan θ + r 1 r 0 ) (r 1 + r 0 ) (L tan θ + r 1 + r 0 ) (r 1 r 0 ) (2.26) This expressin is nly valid as lng as the assumptin f the strain cne angle nt varying thrugh the thickness is true. Very thick plates will have a mre barrel shaped strain frustum [8]. 13

26 With the stiffness K c and the blt stiffness K b, the clamping area can be idealised as a spring system as illustrated in Figure 2.7b. The applied external frce gives a new elngatin f the system: L = ( K b + F a 1 K c1 + 1 K c2 ) 1 = { K C = ( ) } 1 F a = (2.27) K c1 K c2 K b + K C Befre the external lad is applied F c = F pre, where F pre is the pre-tensin. Using the rigid bdy thery t get the applied frce in all blts, the updated clamping frces and frces in the blts can be fund: ( 1 F c = F pre L + 1 ) 1 = F pre LK C (2.28) K c1 K c2 F c = F pre F a K b + K C K C = F pre F b = F pre + LK b = F pre + F a K b K C + 1 F a K C K b + 1 (2.29) (2.30) Clamping Diagram The clamping frces can be described in a diagram as shwn in Figure 2.8. Where F a is the axial external frce n the jint that leads t new defrmatins, d b and d c, in the jint as well as new clamping frces and blt frces. F b and F c is the change f blt and clamping frce. If an axial lad is applied n the jint, a reductin f the clamping frce F c and increase f the blt frce F b is expected. Please refer t Lärbk i Maskinelement [9], fr mre in-depth explanatins f clamping diagrams. Frce F b F c F a F b K b F pre K c F c d b New d b d c Elngatin Figure 2.8: Clamping Diagram New d c 14

27 2.7.2 Effect f Axial Frces n Slip Resistance By intrducing these new stiffnesses it is pssible t frmulate a net change in clamping frces due t the applied axial lad. The blt and clamped material bth experience the prelad frce. The slip resistance withut any axial lad n the clamped area is then assumed t be: R slip,i = 2F pre µ (2.31) The new frces after intrducing the elngatin frm the external frce n the blt and clamped material is calculated as: The new slip resistance is then: F b = F pre + LK b F c = F pre LK C (2.32) The change in slip resistance R slip is: R slip,i = (F b + F c ) µ (2.33) R slip = R slip R slip = (F b + F c 2F pre ) µ = L (K b K C ) µ = F change µ (2.34) Therefre, the net change in clamping and blt frce F change, defined in Eq. (2.34) is calculated as: { } ( ) F a 2Kb F change = L (K b K C ) = L = = 1 F a (2.35) K b + K C K b + K C This means that the cmbined clamping frces decrease depending n the stiffness f the plate cmpared t the blt. In the areas where the axial frce is negative, meaning cmpressive fr the clamping parts, there is a prprtinal increase in slip resistance accrding t the thery. 15

28 3 Finite Element Analysis T validate the previusly develped thery, an FE mdel was created and meshed in the prgram ANSA. The mdel was then slved using MSC s NASTRAN SOL400 with varius chices f parameters. These results were used t cmpare the theretically develped mdel against the behaviur simulated by the FE methd. 3.1 Cntact Pressure In Figure 3.1 fur plts f the cntact nrmal pressure shw sme interesting patterns f the cntact pressure and hw it develps as the jint is laded. The cntact is separated arund the hle f the blts even befre slip, which in a cyclic lading scenari culd lead t fretting wear. (a) Cntact nrmal pressure after prelad step (b) Cntact nrmal pressure at 25% lad applicatin (c) Cntact nrmal pressure just befre slip (d) Cntact nrmal pressure just after slip Figure 3.1: Cntact nrmal pressure f plate experiencing ut-f-plane lad The effects f the ut-f-plane lad can be seen in Figure 3.2 where the defrmatin is scaled by a factr f ne hundred. The gemetrical effects f the ut-f-plane lad, which culd be ne f the explanatins f the increased slip resistance, is clearly shwn. 16

29 Figure 3.2: Defrmatin with scale factr f Slip Detectin Using a nnlinear quasi-static slver means that iteratins during cntact sliding will be made until equilibrium is reached. Slip is thus detected by the slver and can be utput as cntact status, where the gap functin can be pen r clsed, and the cntact in sliding r sticking. Sliding here means that the cntact has been in sliding cnditin during the iteratin, since the slver reaches equilibrium during static cnditins the cntact cannt be in actual sliding during any finished lad step. This can lead t cnvergence issues if the sliding mtin is nt cnstrained. Fr example, a blck sliding n a slpe is nt cnstrained and the equilibrium cnditin fr the sliding blck is nt reachable by the slver unless the blck is cnstrained by a spring r similar, thus decreasing cntact shear frce the further it slides. Slip can be described as a plastic event. The mvement is nn-cnservative and cannt be reversed by delading, detecting glbal slip can therefre be dne by bserving the displacement f say the lading pint. When this pints displacement starts behaving plastically it can be assumed that the jint is slipping. 3.3 Critical Lads Critical lads are picked by hand frm the curves using the assumptin that ttal slip in the jint is reached when the stiffness is heavily reduced. The critical lad is then said t be the ne that is at the nset f slip r just befre slip. The mdels are laded using Rigid Bdy Elements (RBEs) cnnected t a nde, placed ff-centre in the plane and varied ut-f-plane distance, taking a prescribed lad. 17

30 3.3.1 Mdel 1 Figure 3.3a illustrates hw the RBEs are cnnected t the mdel frm the ladpint. The ladpint can then be varied in its psitin t investigate the difference f increased mment arm ut f the plane. Figure 3.3b shws the critical slipping lads depending n the length ut f the plane f the ladpint. Slipping Lad vs Mment Arm Length Slipping Lad [N] st rder element 2nd rder element Thery Arm Length [mm] (a) FE Mdel 1 (b) Slip determined by hand frm lad displacement curves Figure 3.3: Discretised mdel and critical lads fr mdel 1 Figure 3.4 shws the displacements f the individual mdels fr the ladpint in the lading directin. It is pssible t bserve a sticking pint halfway thrugh the slip when mdelling with the secnd rder elements. 1st Order Elements Ladpint Displacement 2nd Order Elements Ladpint Displacement Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Frce [N] Frce [N] (a) Displacement curve fr first rder elements (b) Displacement curve fr secnd rder elements Figure 3.4: Displacements f the ladpint fr each lad step f mdel 1 18

31 3.3.2 Mdel 2 As can be seen frm Figure 3.5a, Mdel 2 has a slightly smaller area f lad applicatin, decreasing the flexural stiffness f the plate and allwing fr mre gemetrical distrtin f the plate. Frm Figure 3.5b the critical lads can clearly be seen t increase cmpared t Mdel 1. Slipping Lad vs Mment Arm Length Slipping Lad [N] st rder element 2nd rder element Thery Arm Length [mm] (a) FE Mdel 2 (b) Slip determined by hand frm lad displacement curves Figure 3.5: Discretised mdel and critical lads fr mdel 2 In Figure 3.6a the sticking halfway thrugh sliding befre the next slip t the blts catching the plate can be seen in the first rder element simulatin. 1st Order Elements Ladpint Displacement 2nd Order Elements Ladpint Displacement Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Frce [N] Frce [N] (a) Displacement curve fr first rder elements (b) Displacement curve fr secnd rder elements Figure 3.6: Displacements f the ladpint fr each lad step f mdel 2 19

32 3.3.3 Mdel 3 Mdel 3 has the smallest area f lad applicatin. This further decreases the stiffness f the plate, leading t even larger gemetrical distrtins. The severe increase in slip resistance is reflected in the plt f the critical lad Figure 3.7b. The increase in slip resistance is then assumed t be due t gemetrical effects mstly. Slipping Lad vs Mment Arm Length Slipping Lad [N] st rder element 2nd rder element Thery Arm Length [mm] (a) FE Mdel 3 (b) Slip determined by hand frm lad displacement curves Figure 3.7: Discretised mdel and critical lads fr mdel 3 In Figure 3.8 the individual displacements are bserved t have lwered stiffnesses due t the decreased flexural stiffness f the plate. 1st Order Elements Ladpint Displacement 2nd Order Elements Ladpint Displacement Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Frce [N] Frce [N] (a) Displacement curve fr first rder elements (b) Displacement curve fr secnd rder elements Figure 3.8: Displacements f the ladpint fr each lad step f mdel 3 20

33 3.3.4 Small Displacements With the large displacements frmulatin turned ff, the results are still similar t large displacements. Figure 3.9b illustrates the similarities. Small Defrmatins Ladpint Displacement Slipping Lad vs Mment Arm Length Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Slipping Lad [N] Large Defrmatins Small Defrmatins Thery Frce [N] Arm Length [mm] (a) Displacement curve fr first rder elements (b) Displacement curve fr secnd rder elements Figure 3.9: Displacements f the ladpint fr each lad step and critical lads fr mdel 3 with large displacement frmulatin and small displacement frmulatin Stiff Plates The Yung s mdulus f the plate material was increased severely, making the plates rigid cmpared t the blts, and the simulatins were re-run. Figure 3.10b reveals that the assumptin f rigid plates fits well with the rigid bdy thery develped. 21

34 Stiff Plates Ladpint Displacement Slipping Lad vs Mment Arm Length Displacement [mm] Mment Arm Length 0 [mm] 50 [mm] 100 [mm] 150 [mm] 200 [mm] 250 [mm] 300 [mm] 350 [mm] 400 [mm] 450 [mm] Slipping Lad [N] Stiff Plates Elastic Plates Thery Frce [N] Arm Length [mm] (a) Displacement curve fr first rder elements (b) Displacement curve fr secnd rder elements Figure 3.10: Displacements f the ladpint fr each lad step and critical lads fr mdel 2 with stiff and elastic plates Observatins Frm the FE results, a clear increase in the slip resistance f the jint with increasing mment arm can be seen. Hwever it is als bserved that the decrease in flexural stiffness als increases the slip resistance f the jint. This leads t the belief that the increasing slip resistance is a gemetrical effect f the plate distrtin. The mdel with stiff plates cnfirms that the plate shape has a big effect n the slip resistance. 3.4 Blt Frces The blt frces during lading are extracted frm the FE mdel. These can then be cmpared t the frces frm the theretical mdel and give a clue abut hw accurate the thery is in estimating the clamping frces in the blted areas. Figure 3.11 shws the theretical blt frces fr a certain lad case cmpared t the FE blt frces fr that same lad case. 22

35 Blt Frces fr 250mm Mment Arm Blt Frces fr 250mm Mment Arm Blt Frce [N] Theretical FEA Thery Critical Lad [N] FEM Critical Lad [N] Blt Frce [N] Theretical FEA Thery Critical Lad [N] FEM Critical Lad [N] Applied Frce [N] Applied Frce [N] (a) Blt frces in jint with 250 [mm] mment arm (b) Blt frces in jint with 250 [mm] mment arm with adjusted lad pint applicatin t match lwer blt pair Figure 3.11: Plt f blt frces frm FE simulatins cmpared t the theretical blt frces during lading As can be seen in Figure 3.11a the thery predicts blt frces in the cmpressive area that are much lwer than the FE results. If the assumptin f the plate rtating abut the centre is changed t rtatin clser t the bttm pair the cmpressive blt pair frces can be matched mre accurately by the thery, as seen in Figure 3.11b. The blt frces in the FE simulatins remain apprximately unchanged in the cmpressive zne. In Figure 3.12 the quadratic behaviur f the blt frces with the increasing lad arm cming frm the FE analysis can be bserved. The thery meanwhile shws cmpletely linear develpment f the blt frces with increasing mment arm. 23

36 Stiff Plate Blt Frces at 20kN Applied Lad Stiff Plate Blt Frces at 20kN Applied Lad Blt Frce [N] FE Result Thery Blt Frce [N] FE Result Thery Mment Arm [mm] Mment Arm [mm] (a) Blt frce in tp right blt at 20 kn applied lad with varying mment arm n an elastic plate Figure 3.12: Plt f blt frces during lading (b) Blt frce in tp right blt at 20 kn applied lad with varying mment arm n a stiff plate A large increase in the blt frces can be seen fr the blts in the tensile area ging well beynd the predicted values f the thery. Due t the elasticity f the plate it can be bserved that frces are nt nly taken in the blted areas, this leads t the cnclusin that the blts in the cmpressive area will nt experience the same cmpressive displacement as the ppsite pair. The blt frces can then be bserved t be fairly cnstant in the cmpressive area f the jint. Meanwhile the clamped material still has t take up the increased cmpressive frce, thus increasing the slip resistance in that area. 24

37 3.5 Real Mdel Simulatin Using a gemetry prvided by Ålö seen in Figure 1.1 and mdified as in Figure 3.13, a simulatin f the glbal slip resistance was made, this ended up prviding further prf that the slip resistance f the jint is imprved with increasing mment arm. (a) Mdel with greater ut-f-plane mment arm Figure 3.13: Mdified mdels (b) Mdel with shrter ut-f-plane mment arm In Figure 3.14 the slipping lad is detected and can be bserved t differ between the tw mdels, where the mdel with increased ut-f-plane mment arm has a small but visible increase in slip resistance. This is then cmpared t the thery develped, where the increase in slip resistance is bserved t be insignificant cmpared t the simulatins. These bservatins fllw the patterns bserved in the previus mdels. 25

38 Subframe Mdel Ladpint Displacement vs Lad Slipping Lad vs Mment Arm Length Displacement [mm] Mment Arm Length 270 [mm] 300 [mm] Slipping Lad [N] Subframe Mdel Thery Frce [N] Arm Length [mm] (a) Displacement curve fr subframe mdel (b) Cmparisn f thery t simulatin Figure 3.14: Displacements f the ladpint fr each lad step f subframe mdel In Figure 3.15 the frces in the blts btained frm the FE simulatin is cmpared t the frces expected by the thery. The rigid bdy mde is crrelated t match the blts in the cmpressive zne, but again the thery fails t capture the extreme blt frces n the tensile znes. Blt Frces fr 300mm Mment Arm Blt Frce [N] Theretical FEA Applied Frce [N] Figure 3.15: Blt frces frm FE simulatin cmpared t thery with crrelated rigid bdy mde t match cmpressive blts 26

39 4 Discussin Plastic Axial Lads Initial experimentatin in an FE envirnment suggests that the slip capacity f a jint increases as the mment arm ut f the plane increases. Hwever, it is nt the authr s recmmendatin t design against slip by mving the lad further frm the jint, as care must be taken s that the blts d nt enter the plastic zne as this culd mean abrupt failure f the jint. The thery develped in this thesis can be used t predict blt frces t fresee such plasticity, but unfrtunately the blt frces predicted in the thery des nt crrelate with the FE results. Blt Head Slip Resistance The blts in the thery develped in this thesis are assumed t carry the slipping lad withut deflecting. Hwever, in a mre accurate setting, this is nt true. At the nset f transverse lading, the blt can be mdelled as a beam with a transverse lad. This means that the blt head frce carrying cnditin is a mix f the slip cnditin and the flexural stiffness f the blt as: { µf b F s min (4.1) δk f where δ is the deflectin f the blt head and K f is the flexural stiffness f the blt. This equatin shws clearly that at the immediate nset f lading there is n cntributin frm the blt head. The blt des nt cntribute t any resistance f the slipping lad until it has started t deflect. Fatigue In Blts The intrduced axial lads frm the ut-f-plane lad culd in a cyclic lading scenari lead t fatigue f the blts, thus leading t jint failure. Using fatigue thery cmbined with the blt frces r the maximum stress in the blt calculated with Appendix B.1 Eq. (B.3) the number f cycles befre failure culd be calculated and that blt can then be remved frm the slip resistance calculatin. 4.1 Future Wrk Further investigatin f the gemetrical shape effect n slip cnditin must be cnducted. A mre accurate methd fr finding the blt lads theretically must be develped as this can prvide insight n hw large effect the ut-f-plane lad has n the axial lads f the blts. A fatigue analysis f the blts shuld als be cnducted as the cyclic axial lading is likely t result in fatigue. An investigatin regarding the pre-tensin lss frm the cyclic axial lads shuld als be cnsidered. An experimental study in a test rig shuld be cnducted t validate the FE mdelling. 27

40 References [1] MSC Sftware. MSC Nastran , Nnlinear User s Guide, SOL 400 (2014). [2] M. Andreassn, MSC Sftware. MSC Nastran SOL400 Wrkshp (2016). [3] MSC Sftware. MSC Nastran Dcumentatin (2014). [4] L. Råde and B. Westergren. Handbk ch frmelsamling i Hållfasthetslära. KTH, (2010). [5] G. L. Kulak. Eccentrically Laded Slip-Resistant Cnnectins (2003). [6] J. E. Shigley. Mechanical Engineering Design. McGraw-Hill Bk Cmpany, Inc, (1963). [7] L. Råde and B. Westergren. Mathematics Handbk fr Science and Engineering. Studentlitteratur, (2007). isbn: [8] K. H. Brwn et al. Guideline fr Blted Jint Design and Analysis: Versin 1.0 (2008). [9] M. Mägi and K. Melkerssn. Lärbk i Maskinelement. EcDev Internatinal AB, (2006). [10] R. Helfrich and I. Pflieger. Simulatin and Optimizatin f Part Cnnectins (2010). [11] P. Wriggers. Cmputatinal Cntact Mechanics, Secnd Editin. Springer, (2006). isbn: [12] M. Shillr, M. Sfnea and J. J. Telega. Mdels and Analysis f Quasistatic Cntact, Variatinal Methds. Springer, (2004). isbn:

41 Appendices 29

42 A Jint Designer Prgram In this chapter the prgram develped t bth cmpare the thery and FE results will be presented. The prgram develped can be used by, fr example, a screw jint designer t find ut hw different blt cnfiguratins and placement f lad affect the critical lad f the jint. A.1 Interactive Script The thery has been develped and is implemented as an interactive script/prgram, where the user can define: Blt psitins Blt pre-tensin Blt and material Yung s mdulus Plate thickness and blt dimensins Frictin cefficient In-plane eccentricity and ut-f-plane eccentricity f the lad In Figure A.1 the utput frm the script is shwn, where the user will get data such as critical lad, centre f rtatin, and clamped area slip resistances. Blt Plt y R_s= 4837 Critical lad= x x Tt Rslip: R_s= 9563 R_s= 4837 R_s= 9563 Lad x Figure A.1: Output frm bltplt script A.2 Parameter Studies The develped script can be used t perfrm parameter studies and see what kind f effects varying parameters have accrding t the develped thery. 30

43 A.2.1 Wrst Blt Mdel Predictins In Sectin 2.5 the methd f determining the first blt t slip yields critical lads that are lwered as the length f the ut-f-plane mment arm is increased. As is shwn this methd is nt gd fr determining jint failure and is nt used further. Figure A.2 displays hw the critical lads when lking at single blts develp with increased mment arms Failure Lad fr Varying Out f Plane Lad Arm Failure Lad [N] Inplane Mment Arm [m] Mment Arm [mm] Figure A.2: Wrst Blt Mdel A.2.2 Varying Clamped Material Stiffness By varying the clamped material s stiffness a plt f the varying slipping lads when increasing the ut-f-plane mment arm can be shwn as illustrated in Figure A.3. 31

44 Slipping Lad fr Varying Material Stiffness Critical Lad [N] e+11 1e+12 1e+13 1e Mment Arm Length [mm] Figure A.3: Varying material stiffness f clamped parts As can be seen, increasing the stiffness f the plates leads t lwered effect frm the ut-f-plane mment arm. B Mtivatin f Rigid Bdy Equatins The equatins used t develp the thery in this thesis has been mtivated with assumptins f small defrmatins. B.1 Area Mment f Inertia The area mment f inertia chsen t calculate the blt frces is presented in Figure B.1. This is under the assumptin f small defrmatins. Intrducing lcal crdinate z fr the blt and glbal crdinate d fr the plate, σ z,i = M z I z σ blt,i = σ z,i + σ mean,i = M z I z z = 0, d = d i + M OP d i n A j d 2 j j σ σ mean,i = M d i n A j d 2 j j d = 0 Figure B.1: Out-f-plane mment effect n the clamped areas 32

45 the stresses intrduced by the mment can be frmulated as: σ mean,i = M d i A i d 2 i σ z,i = M z I z σ blt,i = σ z,i + σ mean,i = M z I z + M OP d i n A j d 2 j j (B.1) (B.2) (B.3) The frce pertaining t the fluctuatin part in Eq. (B.3) (σ z,i ) is then expressed as: F z,i = σ z,i da = z A z M OP z I z = [Antisymmetric] = 0 (B.4) The cntributin F mean,i frm the parallel axis therem is cmputed as: F mean,i = A σ mean,i da = M OP d i A i n A j d 2 j j = M OP d i A i I par (B.5) Where the ttal frce F blt,i in the blt is: F blt,i = F z,i + F mean,i = M OP d i A i I par (B.6) This leads t the assumptin that the nly cntributin t the blt axial frce is frm the parallel axis therem, this is then chsen as the area mment f inertia fr the blted plate. B.2 Out-f-plane Mment Frces The ut-f-plane frces induced by the mment are illustrated in Figure B.2. P 1 y P 2 M OP κ P 3 P 4 Figure B.2: Out-f-plane mment effect n the blts Where an equatin fr mment equilibrium can be frmed as: n P i A i y i = M OP i (B.7) 33

46 A kinematic relatinship between angle κ and strain ɛ i in the blt is given as: Using Hke s law: Insert (B.9) int (B.7): Eliminate κ: ɛ i = κ y i P i = E ɛ i = E κ y i n E κ A i yi 2 = M OP i κ = M OP I E, where I = The mean pressure in the blt is then expressed as: P i = M OP y i I par Where the axial frce F OP frm the ut-f-plane mment is: n j A j d 2 j F OP = P i A i = M OP r i A i I par (B.8) (B.9) (B.10) (B.11) (B.12) (B.13) B.3 In-plane Mment Frces The kinematics fr in-plane mment equilibrium is illustrated in Figure B.3. T 1 T 2 r 2 M IP r 1 r 3 r 4 r 5 T 4 T 3 T 5 Figure B.3: In-plane mment effect n the clamped areas Where an equatin fr mment equilibrium can be frmed as: n T i A i r i = M IP i (B.14) 34