Plastic deformation in flat-section band clamps

Transcription

1 93 Plastic deformatio i flat-sectio bad clamps K Shoghi, S M Barras, ad H V Rao Departmet of Mechaical Egieerig, Uiversity of Huddersfield, Huddersfield, UK The mauscript was received o 20 May 2004 ad was accepted after revisio for publicatio o 22 October DOI: / X8388 Abstract: Previous work o the elastic deformatio of flat-sectio bad clamps has bee exteded to accout for plastic deformatio of the bad material. Both fiite elemet aalysis usig a multiliear plastic model ad a classical approach usig a cotiuous plastic equatio are reported. The excellet agreemet betwee the results of these two approaches provides a high level of cofidece i the models. The fiite elemet approach was foud to be extremely time cosumig owig to the combiatio of chagig cotact o-liearity ad material oliearity. The classical approach provided a much faster solutio method whe used with the iteratio schemes proposed here. The classical method of aalysis was used to determie the respose to a rage of coditios. This aalysis showed that there might be some advatage i drivig flat-sectio bad clamps ito the partially plastic state. I this state the coefficiet of frictio has far less impact o the variatio i circumferetial stress, ad hece radial clampig force, aroud the bad. Keywords: stress, flat-sectio bad clamp, cotact with frictio, rigid cylider, plastic deformatio, material o-liearity, fiite elemet model, coefficiet of frictio 1 INTRODUCTION Flat bad clamps are widely used for coectig flexible tubes or hoses to comparatively rigid ducts or pipes. They cosist of a simple circular bad of material carryig two truio straps, as show i Fig. 1. The bad is tighteed usig a T-bolt coectig the two truios. The purpose of the flat-sectio bad clamp is to provide a pressure seal betwee the compoets by supplyig the required radial pressure o the flexible compoet of the joit. Bad clamps have bee available for may years, ad developmets to try to esure that the radial load is evely distributed have take place over a similar period. These iclude overlappig the eds of the bad, as suggested by Joes [1], ad icludig iserts i the area uder the T-bolt, as advocated by Schaub [2]. Recetly, Shoghi et al. [3] have described the elastic behaviour of such clamps ad demostrated that, owig to frictioal effects, it is ulikely that the radial load will be evely distributed. Correspodig author: Borgwarer Turbo Systems, Roysdale Way, Euroway Idustrial Estate, Bradford, West Yorkshire BD4 6SE, UK. I demadig applicatios ad durig maiteace, a bad clamp may well be tighteed beyod the origial desig load. Typically, ultimate failure will occur i the T-bolt rather tha the flat bad (see referece [4]). However, before this failure poit is reached, it is likely that, owig to the o-uiform stress distributio aroud the clamp, plastic deformatio will occur i some areas. While there will be o visible effect from this deformatio, it could well compromise operatioal performace. There is therefore a eed to uderstad this partially plastic state. Although fiite elemet methods could be employed to fid the relatioship betwee stress ad displacemet beyod the yield poit, the bad-clamp problem is highly o-liear owig to the cotact coditio. Addig material o-liearity to the problem makes obtaiig a coverged solutio difficult ad time cosumig. Cosequetly, the preset paper sets out a theory applicable to a wide rage of flat-sectio bad clamps takig accout of material o-liearity. 2 DEVELOPMENT OF MODELS The hoses joied by flat-sectio clamps are typically made from flexible materials, while the ducts or C09504 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece

2 94 K Shoghi, S M Barras, ad H V Rao i the elastic regio will be as give by Shoghi et al. [3]. The relatioship betwee the clampig load, F b, ad hoop stress, s a, at agle a (see Fig. 2) is give by Shoghi et al. [3] as s a ¼ F b exp½ma ð bþš wt ð1þ Fig. 1 Flat-sectio bad clamp compoets pipes to which they are beig coected to are comparatively rigid. I the aalyses preseted here, the compoets beig joied are assumed to be rigid. The flexible ature of hoses, ducts, etc., may be take ito accout by makig the fial closed radius of the bad, R (see Fig. 2), equal to the compressed outer radius of such flexible compoets. 2.1 Theoretical stress displacemet relatioship As with the elastic aalysis previously preseted [3], it is assumed that stress withi the bad is uiform both across the width ad through the thickess. The aalysis of the bad ca the be divided ito two parts, a elastic portio govered by Hooke s law ad a o-liear regio. The stress distributio This relatioship was established by cosiderig force equilibrium ad is therefore applicable to both the elastic ad plastic regios. However, F b is difficult to cotrol or measure. It is therefore ecessary to relate the hoop stress to the more readily cotrollable circumferetial displacemet, d. From equatio (1) it ca be see that the largest stress will occur at the gap. Hece, yieldig ad plastic deformatio will iitiate at this poit ad gradually propagate towards the back of the clamp (a ¼ 0). The boudary betwee the elastic ad plastic regios will occur at some agle, h. At this boudary the value of stress s a give by equatio (1) will equal the yield stress, s Y. Hece, from equatio (1) s Y ¼ F b exp½mðh bþš wt The boudary agle, h, is therefore give by h ¼ b 1 m l F b wts Y ð2þ ð3þ The elastic circumferetial deformatio of the bad was foud previously by Shoghi et al. [3]. For a elastic regio 0, a, h, this deformatio will be d E ¼ RF b Ewt Hece ð h 0 exp½mða bþšda d E ¼ RF b½expðmhþ 1Š Ewtm expðmbþ ð4þ The relatioship betwee stress ad displacemet, d p, for the o-liear regio should be determied takig ito accout the o-liearity due to the material. From Fig. 3 the strai is give by 1 ¼ dd p R da ð5þ Fig. 2 Elastic plastic aalysis of the flat sectio bad clamp where dd p is the extesio of a small segmet, R da, of the bad. Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece C09504 # IMechE 2005

3 Plastic deformatio i flat-sectio bad clamps 95 Hece F 1= ð b b d p ¼ R exp ma da Awt expðmbþ h ð9þ Thus, the value of the plastic displacemet, d p, ca be expressed as d p ¼ R m F 1= b 1 exp Awt h m i ðh bþ o ð10þ ad the total displacemet, d, is give by Fig. 3 Deformatio of a bad segmet As Hill [5] describes, there are a umber of models available to simulate the o-liear behaviour of metals. Oe of the simplest of these is the power law d ¼ d E þ d p Hece d ¼ R F b ½expðmhÞ 1Š þ F 1= b m Ewt expðmbþ Awt h m io 1 exp ðh bþ ð11þ s ¼ C þ A1 The difficulty i usig this model for the whole stress strai curve is that it teds to uderestimate stress at low levels of strai ad overestimates stress for larger strais. However, the slightly simpler model s ¼ A1 ca give quite a reasoable fit to the plastic regio aloe. I the implemetatio here, this equatio is used i combiatio with a liear model for the elastic regio. I order to determie the costats A ad, two poits o the stress strai curve are selected such that the strai values betwee these two poits covers the required rage of strai. A pair of simultaeous equatios for A ad is thereby geerated. Combiig equatios (5) ad (7) allows the stress at agle a withi the plastic regio to be related to the plastic deformatio at that poit s a 1=¼ dd p A R da Combiig equatios (1) ad (8) gives dd p ¼ F b exp½mða bš 1= R da Awt ð6þ ð7þ ð8þ 2.2 Method for calculatig the stress distributio i the bad for a give displacemet Calculatio of the stress distributio i the bad at a give displacemet usig the elastic plastic model is complex ad requires a iterative procedure. A suitable procedure is give by the flow chart show i Fig Agular displacemet aroud the bad with plasticity I order to determie the displacemet at differet agular locatios o the bad it is ecessary to establish a expressio for the icremetal plastic displacemet, d ip, at agle a p. Usig equatio (9) gives d ip ¼ R F b expð mbþ 1= ð ap Awt h exp ma da Hece, the icremetal plastic deformatio is give by d ip ¼ R F b expð mbþ 1= Awt h m exp ma p exp mh i This is valid for a p. h. ð12þ C09504 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece

4 96 K Shoghi, S M Barras, ad H V Rao allows cotact pressure to be correctly calculated by itegratig over the face of the elemet. Hece, the usual difficulty associated with cotact ad higher-order elemets is avoided. Surface-to-surface elemets were used for this aalysis. These elemets are well suited for this type of applicatio where a sigificat amout of slidig ad frictio is ivolved. These cotact elemets ca be used with both lower ad higher-order cotiuum elemets withi the simulated model. Symmetry of the flat-sectio bad about two plaes was used to reduce the size of the model ad to allow costraits to be applied o the plaes of symmetry show i Fig. 5. The clamp was loaded by applyig a pressure to the ed face of the clamp equivalet to the circumferetial force applied by the T-bolt. The bad was also costraied alog the edge of the sectio, as show, to elimiate potetial rigid body displacemet. 2.5 Modellig material o-liearity Fig. 4 Method used to calculate stress distributio i the bad The total icremetal displacemet, d T, at agle a p is give by d T ¼ d ip þ d E ð13þ I the developmet of the theoretical model of the bad clamp described i sectio 2.1, a o-liear material model of a form similar to those described by Hill [5] was utilized. Oe difficulty i usig this type of model is i matchig it to the observed yield stress ad the elastic behaviour. For materials that do ot exhibit a distict yield poit, a percetage proof stress is ormally used to idicate the oset of plastic strai ad replaces the yield stress i may desig formulae. I this work a 0.2 per cet proof stress, s 0.2, has bee used. For the aalysis described here, the yield stress must occur at the itersectio of the liear elastic regio ad the plastic stress strai curve approximated by equatio (7). 2.4 Fiite elemet aalysis A model of the flat-sectio bad iteractig with a rigid cylider was costructed usig threedimesioal elemets with a quadratic iterpolatio polyomial. The reaso for usig a quadratic elemet was to esure that the elemets took up the curved shape of the bad clamp ad cylider. This criterio is best explaied by imagiig a bicycle chai wrapped aroud a rigid cylider. The straight chai liks will prevet full cotact betwee the chai ad the cylider, ad hece reduce the cotact area. A similar effect would have bee observed i the case here if liear elemets had bee used. Ciabattoi (D. Ciabattoi, 2000, persoal commuicatio) cofirmed that, i the fiite elemet code used for this problem, cotact is detected at the Gaussia itegratio poits. This Fig. 5 Fiite elemet model Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece C09504 # IMechE 2005

5 Plastic deformatio i flat-sectio bad clamps 97 Hece, at the yield poit s Y E ¼ s Y 1= A Solvig for yield stress from the above equatio gives s Y ¼ ð1= 1Þ E ð14þ A Usig equatio (14), the modelled value of yield stress, s Y, ca be calculated. Modellig the material behaviour ow ivolves the stadard process of determiig the Youg s modulus (see, for example, referece [6]), establishig a suitable fit withi the plastic regio, ad matchig these two models at a sesible yield poit. The iterative process illustrated by the flow chart show i Fig. 6 was used to achieve this. Withi the fiite elemet software a multiliear isotropic hardeig model was used rather tha the more complex sigle curve i the theoretical model. The o-liear material behaviour may be defied usig the data obtaied from a tesile test as a sequece of up to 20 stress strai data poits withi the plastic regio. Realistically, it is usually faster ad more robust to use a simplified multiliear model with five or six slopes. It is importat that the slope (s 1 /1 1 ) for the first poit of etry correspods to the elastic modulus give i the material properties. Here, s 1 ad 1 1 refer to the stress ad correspodig strai values for the first poit. No part of the curve segmet may have a slope larger tha the elastic modulus. Negatives slopes are ot recommeded ad could cause covergece problems. The software assumes elastic perfectly plastic behaviour for the values past the ed-poit defied by the table. 3 MODEL COMPARISON 3.1 Geometry ad material data To ivestigate the performace of the fiite elemet ad theoretical models, they were used to ivestigate the behaviour of a sample bad clamp. The dimesios of this clamp are give i Table 1. The stress strai curve show i Fig. 7 was determied by carryig out a tesile test o a sample of bad material. The process show i Fig. 6 was used to determie the material model show i Fig. 7 ad Table 2. To defie the material model for the fiite elemet simulatio, the iitial yield stress show i Table 2 was used alog with a additioal four poits take from the tesile test data. These poits alog with the multiliear material model are also show i Fig. 7. The elastic modulus used i the fiite elemet model was 227 kn/mm 2 ad Poisso s ratio was Discussio of the results for the fiite elemet model The fiite elemet model described i sectio 2.4 was aalysed with a ed load of 16.1 kn applied as a pressure load to the cross-sectioal area ear the ope gap. The maximum correspodig vo Table 1 Properties of the flatsectio bad clamp Fig. 6 Method used to determie the theoretical material model Width, w Bad thickess, t Iitial radius, R Subteded agle of half the bad, b mm 1.22 mm 59.5 mm 1628 C09504 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece

6 98 K Shoghi, S M Barras, ad H V Rao Fig. 7 Stress strai curve Mises stress i the flat-sectio bad is MPa ad occurs ear the gap. A coefficiet of frictio, m, of 0.3 was used for this model. The stress plot is show i Fig. 8. As a quality check, the circumferetial stresses predicted by the fiite elemet model at the ed of the bad were compared with the applied stress of 700 MPa. The stress from the model was 682 MPa, idicatig a error due to stress field approximatio of less tha 3 per cet. The differece betwee the vo Mises equivalet stress ad the circumferetial stress is due to the presece of the direct radial stress due to cotact ad shear stresses resultig from frictio betwee the bad ad the flage. I the fiite elemet model it is ot straightforward to idetify the boudary betwee the elastic ad plastic regios accurately. This is defied as agle h. This positio ca be show i the fiite elemet aalysis results by settig the yield stress as the miimum stress to plot, as show i Fig. 8. The value of agle h ca the be determied from the positio of the odes o or close to this boudary. located o the model. This is show i Fig. 9 alog with the results predicted by the theory preseted i sectio 2 ad the results of the liear aalysis of the same problem as preseted by Shoghi [3]. The excellet correlatio betwee these models is immediately evidet. The effect of material oliearity is also clearly show. As expected, i the elastic regio the elastic ad elastic plastic models 3.3 Compariso of results Havig idetified the boudary betwee the elastic ad plastic regios, the plastic displacemet ca be Table 2 Material properties of the flat-sectio bad clamp Costat, A Costat, Elastic modulus, E 227 kn/mm 2 Yield stress, s Y 525 N/mm 2 Fig. 8 Plastic stress plot Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece C09504 # IMechE 2005

7 Plastic deformatio i flat-sectio bad clamps 99 Fig. 9 Agular displacemet for elastic ad plastic regios Fig. 10 Effect of material o-liearity o total displacemet C09504 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece

8 100 K Shoghi, S M Barras, ad H V Rao Table 3 Summary of the results for cotact with frictio ad material o-liearity Coefficiet of frictio, m Method of calculatio FEA Theory FEA Theory FEA Theory Displacemet, d (mm) Differece (%) Elastic plastic boudary agle, h (deg) Differece (%) give idetical results. However, i the plastic regio the o-liear model produces much larger displacemets. The relatioship betwee force ad total displacemet with two differet coefficiets of frictio is show i Fig. 10. The graph demostrates good correlatio betwee the theory developed i sectio 2 ad the fiite elemet model with material oliearity. It is worth otig that, as the force is icreased, the correlatio betwee theory ad fiite elemet model has slightly deteriorated. This is due to the limits iheret i the material model defied by equatio (7) ad Table 2. This limitatio ca also be see i Fig. 7 where, for higher strais, the stress predictio deviates from the test results. However, correctios could be made by obtaiig a differet rage for equatio (7) to cater for higher loads. This ca be achieved by chagig the costats A ad. A summary of the results from the fiite elemet ad theoretical models for a clampig load F b ¼ 16 kn is give i Table 3. The correlatio betwee the results obtaied usig equatios (3) ad (11) ad the fiite elemet method is excellet. Table 3 shows a larger differece for a coefficiet of frictio of 0.3 compared with This is due to stress cocetratio withi a smaller plastic regio ear the gap ad ca be corrected by refiig the mesh desity i this area. Verificatio of o-liear results is more complex tha verificatio of liear results, because, by their ature, o-liear results ted to be harder to predict. However, the excellet correlatio betwee the theoretical ad fiite elemet results preseted here provides cofidece i both calculatio methods. 4 NON-LINEAR EFFECTS AND BAND PERFORMANCE 4.1 Relatioship betwee stress ad displacemet I order to examie the effect of the o-liear material model, the graph of hoop stress agaist displacemet show i Fig. 11 for the flat-sectio bad clamp aalysed i sectio 3 was plotted. Equatios (1), (4), ad (11) were used to predict the stress for the agles a ¼ 162, a ¼ 90, ad a ¼ 0 for the bad described i Tables 1 ad 2. A coefficiet of frictio of 0.15 was assumed. It should be oted Fig. 11 Hoop stress at agle a for liear ad o-liear material model Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece C09504 # IMechE 2005

9 Plastic deformatio i flat-sectio bad clamps 101 Fig. 12 Stress distributio aroud the clamp here that the displacemet ad stress due to iitial bedig of the bad clamp has bee omitted for clarity. It ca be see that, while the bad clamp is fully elastic, the graph remais liear. A chage i the slope of the graph shows the start of yieldig. As the elastic plastic iterface, h, moves aroud the bad clamp towards a ¼ 08, the displacemet icreases rapidly for small icreases i hoop stress. It ca be observed here that, sice the total circumferetial displacemet is a fuctio of the strai aroud the whole clamp, yieldig at oe poit i the clamp makes the stress displacemet relatioship o-liear at all other poits i the clamp. Hece, the o-liearity observed is due to both the material o-liearity ad the developmet of the plastic regio. 4.2 Stress distributio Usig the procedure described i the flow chart, give i Fig. 4, the graph of the hoop stress at differet levels of frictio with the o-liear material model was determied at differet values of agle aroud the bad for a total displacemet value d T ¼ 0.6 mm. This level of displacemet geerated plastic deformatio i the clamp across the full rage of frictio coefficiets. All other properties of the flat-sectio bad were as give i Tables 1 ad 2. The same calculatio was carried out usig the purely elastic model of bad behaviour previously preseted by Shoghi et al. [3]. The results of these aalyses are show i Fig. 12. As expected, the elastic model predicts higher stresses i order to geerate the same level of displacemet. It is also observed that, at low levels of frictio, the plastic model gives a larger variatio i stress aroud the clamp. However, at higher levels of frictio this situatio is reversed, with the o-liear model givig more eve stresses. These variatios i stress are importat sice the radial clampig force is directly proportioal to the circumferetial stress. It ca also be see that the o-liear model predicts a much smaller effect due to variatios i the coefficiet of frictio. This illustrates a possible advatage i drivig this type of clamp ito the plastic regio sice the coefficiet of frictio is probably the least cotrollable variable i this type of applicatio ad ca vary substatially depedig o the material beig clamped. 5 CONCLUSIONS A theoretical model has bee developed to predict the deformatio i flat-sectio bad clamps where the material has bee loaded ito the plastic regio. A method for determiig the elastic plastic stress distributio for a applied displacemet has also bee proposed. This classical approach has bee validated agaist the results of fiite elemet aalyses of the problem, with differeces betwee the two approaches beig less tha 3 per cet. The larger differeces occur at higher levels of frictio owig to the difficulty i modellig the larger cocetratio of stress towards the bad T-bolt. The circumferetial stress, ad hece the radial clampig load, i partially plastic bad clamps shows sigificatly less variatio due to chages i the coefficiet of frictio tha the equivalet elastic bad. However, at lower coefficiets of frictio the partially plastic bad exhibits more stress variatio aroud the clamp. REFERENCES 1 Joes, W. G. Hose clamp for thi wall tubes. US Pat , Schaub, E. T-bolt hose clamp. US Pat , C09504 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece

10 102 K Shoghi, S M Barras, ad H V Rao 3 Shoghi, K., Rao, H. V., ad Barras, S. M. Stress i flat sectio bad clamp. Proc. Ist Mech. Egrs, Part C: J. Mechaical Egieerig Sciece, 2003, 217, Shoghi, K. Stress ad strai aalysis of V-sectio bad clamps. PhD thesis, Uiversity of Huddersfield, Hill, R. The Mathematical Theory of Plasticity, 1950, pp (Oxford Uiversity Press, Lodo). 6 Dea, G. D., Loveday, M. S., Cooper, P. M., Read, B. E., Roebuck, B., ad Morrel, R. Aspects of modulus measuremet. I Materials Metrology ad Stadards for Stadard Performace (Eds B. G. Dyso, M. S. Loveday, ad M. G. Gee), 1995, Ch. 8, pp (Chapma ad Hall, Lodo). APPENDIX Notatio A E F f F a F b costat elastic modulus (GPa) frictio force (N) circumferetial force at agle a (N) clamp force (secod stage) (N) R t W costat radius of the bad (m) thickess of the bad (m) width of the bad (m) a cotact agle for the bad (rad) a p cotact agle for the bad for the plastic regio (rad) b subteded agle of half the flat bad (rad) d total displacemet (m) d E elastic displacemet (m) d ip icremetal plastic displacemet (m) d p plastic displacemet (m) d T total icremetal displacemet at agle a p (m) 1 strai h elastic plastic boudary agle (rad) u T total agular displacemet (rad) m coefficiet of frictio betwee the bad ad the rigid cylider s Y yield stress (MPa) s a hoop stress at agle a (MPa) 0.2 per cet proof stress (MPa) s 0.2 Proc. IMechE Vol. 219 Part C: J. Mechaical Egieerig Sciece C09504 # IMechE 2005