filete sem concentração de tensão

7/28/2019 Filete sem concentrao de tenso

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F I L L E T S W I T H O U T S T R E S S C O N C E N T R A T I O NR. LANSARD, L ab or at oi re s de l a socie/tG de s Autom obiles Peugeot,

Sochaux (Doubs), France.

ABSTRACTIn a circ ular fi l let , however la rge the radiusis , the re is always a s tr es s concentration. It

i s possible to design prog ressive curvatur efi l lets , in which stress is constant along theprofile, so there i s no mo re chance of fail urein the fillet than anywhere else. This papergives the profile of such fi ll et s in the ca se ofplates in tension or bending.

LIST OF SYMBOLSo = width of channel,

i i i i T i A i F L 0 w VEL O C IT Yv, = FLOW VELOCITY IN CONTRACTED SECTION '

ceptable, the par t may be too lar ge e lsewhere.Le t us suppose that a profile of a fill et can bedesigned without s t r e ss concentration, thenthe re i s no mo re chance of a failur e in thefill et than anywher e els e, and no weight needbe added. This i s the so-called "optimumfi l le tw. ~ a u d ~n the U.S.A. and ~ a l e t inFra nc e have published pape rs on this subjectand investigated progressive curvature fi l lets .

The purpose of t his paper i s to study ex-periment ally the st r e s s concentration of so mefil let profiles derived from hydrodynamicproblems.

DETERMINATION OF THEOBTiMUivi FILLET

b = 2 AC = width of channel - width of or ifi ce , The optimum fil let is defined by the con-8 = angle of tangen t at any point of the pr ofi le, dition of const ant s t r e s s along the profile. ByD = la rg e width of the plat e o r channel, fixing the s t r e s s conditions along the contourd = sma ll width of the pla te o r channel, (load conditions) and transfor ming this prob-6 = width of the orifice, lem by a conformal transformation, analogous

to the hodograph tr ansf orm, it could be pos-sib le to find a field function solution of thi stransformed problem where the contour is

INTRODUCTION com plet ely known, usin g the rel axati on method.Another a ppr oach would cons ist in definingIn a mechanical s tru ctur e, a fillet is often a an exact mathematical correspondence be-

weak point. The stress is la rg er than the tween the elas tic field and a hydrau linominal value calc ulated by cla ssica l formula e 8the solution of t he la tt er being known .of strengt h of mate rial s. Seve ral auth ors l* Thi s method was used by Salet in his studyhave given tables and c ha rt s for s t re ss con- of th e optimum fi ll et fo r a hollow shaf t in tor -sion. The str es s field is represented by a

When the s t r e s s in the fillet is made ac- hydrodynamic field, where velocity co rre -sponds to shea r. The optimum profile cor re-spon ds to the prof ile of a jet. If the influencePres ent ed a t the Annual Meeting of the Society for E xperi- of gravity is neglected, the pre ssu re do ng themental Str ess Analysis in Philadelphia, Pa., September, 1954. sur fac e of a fr ee jet is constant, and therefor e* Superiors pertain to refe renc es listed a t the end of the the velocity is constant- The ho do gr a~ h s acir cle 4. The problem i s solved by studying


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S . E . S . A . P R O C E E D I N G S V O L . XI11 N O . 1

- D

F IG. I . PLATE IN TENSION.

F IG. 2 . TWO-DIMENSIONAL FLOW THROUGH A SLOT( OR THREE-D IMENS IONAL FLOW THROUGH ACIRCULAR ORIFICE ).

FIG. 4 . LOADING MACHINE.

I t -5 ?7T' in electrostatics, constant e lec tr i c f ield su r-fac es a re some tim es used in the design ofcondenser plates, avoiding dielectric failures.Fel ici studied this problem by conformal map-9' ping and analogy with hydrodynamic flow 5.----! C2 4" P L A T E IN T E NS IO N

F I G . 3. MODEL WITH TENSIO N LOADING DEVIC E.

Consider a plate (Fig. 1) subjected to puretension. The proble m is to determine the op-timum fillet, from C to E . The s t r e ss i s sup-posed uniform in the extreme sections.An intuitive "anal ogyn is the two-dimensionalhydraulic flow through a slot AB (Fig. 2) . Thechannel width is C D . Ther e i s a contraction ofthe jet, and at a sufficient dist anc e from theslot, the jet width can be considered as con-stant. In the extreme sections, the velocity isexperimentally the hodographic tra nsfo rmed uniform.field, by the electro lytic tank method. Fr om The hydrodynamic problem was solved byelectr ical measurements, i t is possible to cal- B. Betz and E . Peter sohn using the hodographculat e the profile coordinates. method, and the profile coordinates x and YAnother analogy lead s to a si mi la r problem - a r e :


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F I L L E T S W I T H O U T S T R

where L = Napierian logarithm,o = wall width,v, = ra ti o of ini tia l velocity to velocityin section E F ,8 = angle of pr ofi le tangent at point

( X , Y ).The s lo t width is given by the relation:

E S S C O N C E N T R A T I O N

to see if it could be used in ind ust ria l des ig nsa s an approximation of the optimum fillet.Fou r value s of the width ra ti o K were stud-ied ( K is the quantity v, in the X Y formulae).K = 0.819,0.6 ,0.4,0.2.

x and y values were calculated for reg ula r-ly spaced values of coo 0 . As it was better formachining facilities to have regula rly spacedvalues of X , they were derived by graphicalinterpolation on a large scale drawing.Tables I to IV give the values of the co ordi -nates, assuming o = 1.STRESS ANALYSIS O F THE PROFILE

Without research and without defining anyexa ct and mathemati cal analogy between thest re s s field and the hydrodynamic field, theprofile was tested a s a fillet profile, in orderA photoelastic st re ss analysis was made ofthe profile. A model was made of h' Catalin.Fig. 3 shows the model and the loading sy ste m.

F I G . 5 . F R I N G E P A T T E R N S FOR P L A T E S I N T E N S I O N .THE BO TTO M L IN E OF EACH PICTURE IS THES Y MME TRY A X IS O F THE P LATE .


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S . E . S . A . P R O C E E D I N G S V O L . X I 1 1 N O . 1

TABLE I.PLATE I N TENSION.

K = 0 .2

TABLE E.P L A T E I N T E N S I O N .

K = 0.4

The st r es s distribution in the PP' and OQ' sec - machined on a pantograph milling machine,tions was uniform within 5%. The loading using a large scale drawing a s a master . Thismachine wa s the sam e a s in Fig. 4. method proved to be inconvenient for the pr es -In pre limi nar y tes ts in Franc e, the model ent study, and the profi le was machined directlywas hand-made fr om a ste el ma st er which was in Catalin, ste p by step, on a conventional


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F I L L E T S W I T HO U T S T R E S S C O N C E N T R A T I O N

TABLE E!.PLATE IN TENSION.

K = 0.6

TABLE IV.PLATE IN TENSION.

0.10820.10840.10860.11010.0900

ile wa

Y3.02883.03000.03120.03240.03360.03460.0358

Fig. 5 shows the frin ge patte rns. Along onefring e the difference of principal st re ss es isconstant.

X

0.13200.13600.14000.15000.16000.17000.1800

precision milling mafinished by hand to remo ve the step s. Usualcar e was taken in machining, a s needed forphotoelastic studies. Four model s were tested,having the val ues of K previously given.

0.0368 /0.03780.03880.03980.04080.04180.04260.04360.04440.04540.04620.04700.04780.04860.0494 '0.05020.05100.05160.05240.05320.05380.05520.05660.05780.05900.06020.06140.06260.06380.06480.06580.06660.0676

line.

0.19000.20000.21000.22000.23000.24000.25000.26000.2700d.28000.29000.30000.31000.32000.33000.34000.36000.38000.40000.42000.44000.46000.48000.52000.56000.60000.64000.68000.72000.76000.8000AC =

The prc


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S . E . S .A . P R O C E E D I N G S V O L . XI11 N O. 1TABLE V.

PLATE IN BENDING.K = 0 . 2 0

TABLE VI.PLATE IN BENDING.

K = 0 . 4 0

FIG. 6. PLATE IN BENDING.

t

F I G . 7. MODEL WI TH BENDING LOADING DEVICE.

I t is clea r that, along a fre e edge, as a,vanishes, a, k . If the fillet i s "optimum",the load can be adjusted for having a fringeall along the fillet profile.If one exa min es the pat ter ns of Fi g. 5, i t canbe seen that this condition is not exac tly real -ized. The st re ss in the sharp pa rt of the filletis about 30% le ss than the nominal st re ss . Theinvestigated fillet is not the optimum fillet, butit i s in the direction of security. It se em sthat a better approximation could be found,using fillets with a curvature varying mor erapidly.

PLATE IN BENDINGConsider the plate (Fig. 6 ) subjected to purebending. The probl em i s to det erm ine the op-

timu m fillet. In a sim ila r way a s for tension,consijer the "analogy" with an axially sym-met ric al flow fro m a ci rcul ar orifice withsliameter AB (Fig. 2).

No rigorous analogy was sea rch ed for be-tween these two pro ble ms. The pur pos e of thestujy was to test the jet profile as a fillet


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L E T S W J T R A TTA B LE V I I I .

P LA TE IN B E NDING .K = 0.75

TABLE VI I .P L A T E I N B E N D IN G .

Tables V to VIII give the profile coordinates;the large width cD i s supposed equal t o unity.STRESS ANALYSIS O F THE PROFILE

The model machining was the s am e a s forthe tension study. Fig. 7 shows the loadingdevic e for ci rc ul ar bending. Fig. 4 sho;vs theloadi ng machine. Fig. 8 se ts forth the fringepattern s. It can be seen that the investigatedprofile is not too far from being the optimumfil let - within about 5%.

profile for plates in bending.The h 3rodynamic problem was stuJied byTrefftzJ by the relaxation method, by Kretz-sc hm er who extrapolated the re su lt s of Betzand Peterso hn's two - dimensional analysis,and by Rou se and Abul Fetouh 10 who used twodifferent approaches: the relaxation methodand the electr olytic tank method. Pro fil e for-mulae a re not given because "no three-dimen-sional counterpart has yet been devised forthe powerful two-d imens ional tool of con for maltra nsf orm ati on" 10. Rous e and Abul Fetouhgave their r es ul ts a s tables of the coordinatesr and Y , for four values ofthe ratio of 8 to D.These profi les were tested a s fi l let profi les.The investigated ratios were:

K = 0.75,0.58,0.40,0.2 0.

CONCLUSIONThe investigated profiles , a s defined byTables I to VIII or by Betz and Petersohn'sformul ae, can be considered a s giving a goodapproximation of the optimum fillet fo r indus-

tria l needs.There are some reasons to think that thebending prof ile would not be too bad in theca se of a cir cula r shaft in pure tension o rpur e torsion. It would be an intere sting in-vestigation tos tudy this problem by the frozenfring e pat ter ns technique, o r by wire st ra ingages.


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S . E . S . A . P R O C E E D I N G S V O L . XI11 NO. 1

FIG. 8 . FRINGE PATTERNS FOR PLATES IN BENDING.THE BOTTOM LINE OF EACH PICTURE IS THESYMMETRY AXIS OF T H E P L A T E .

ACKNOWLEDGEMENTSThe author is deeply indebted to Prof. W. M. Mu rr ay for his helpful suggestions, and continuedencouragement. Valuable assistance was received from Mr. P. K Stein, of the ExperimentalSt re ss Analysis Laborato ry, Massachusetts Institute of Technology. This work was ca rr ie d out in1951 with the finan cial as sis tan ce of the M IT Foreig n Students Summer Project. The authorwishes to express his appreciation to all the individuals who gave him the opportunity of workingfor some months in American laboratories.

REFERENCES1. L i p s o n , C., Noll , G. C., an d Clo ck , L . S .; S t re s s an d Pa t t e r n s , Jo u r n a l of Ap p l ied Ph y s ic s , Vo l. 1 2 , Au g . 1 9 4 1.

S t r e n g t h of M a n u f a c t u r e d P a r t s , M c G r a w H i l l B o o k C o m p a n y , 7 . B e t z , B., a n d P e t e r s o h n , E .; A n w en d un g d e r T h e o r i e d e rN e w Y o r k , F i r s t E d i t i o n, 1 9 5 0 . f r e i e n S t r a h l e n , I n g e n i e u r A r c h i v , V ol . 2 , 1 9 3 1 , p. 1 9 0 .2. B a u d; F i l l e t P r o f i l e s f o r C o n s t an t S t r e s s , P ro d u c t E n - 8. T r e f f t z , E. V.; U b e r d i e K o n t r a k t i o n k r e i s f o r r n i g e rg in ee r i n g , Ap r i l 1 93 4 , p . 1 3 3 . f l i i s s i g k e i t s s t r a h l e n , Z e i t s c h r i f t fi ir M a t h e m a t i k u n d P h y s i k ,3 . Sa le t , G.; F o r m e s r a t i o n n e l l e s d e s c o n g bs d e r a c c o r d e - Vol. 64, 191 7, p. 34 .m e n t d a n s l e s a r b r e s d e r 6 vo l u t io n t r a v a i l l a n t e n t o r s i o n, 9. K r e t z s c h m e r , F.; S t r o m u n g s f o r m u n d D u r c h f l u s s z a h lBu lle t in de l1A.T.M.A., 1946, pp . 109-124. d e r M e s s d r o s s e l n , V e r e i n D e u t s c h e r I ng e n i e ur e , N o. 3 8 1 , 1 93 6 .4. L a m b , S i r H. ; H y d r o d y n a m i c s , D o v e r P u b l i c a t i o n s , N e w 1 0 . R o u s e , H ., a n d A b d e l H a d i A bu l F e t o u h ; C h a r a c t e r i s t i c sY o r k , S i x t h E d i t i o n, 1 9 4 5. o f I r r o t a t i o n a l F l o w T h r o u g h A x i al l y S y m m e t r i c O r i f i c e s ,5 . d l i c i , N .; L e s s u r f a c e s c h a m p 6 1 e c t ri q u e c o n s t a n t , J o u r n a l of A p p li e d M e c h a n i c s , V ol . 1 7 , N o. 4 , D e c . 1 9 5 0 , p p.R e v u e & n k r a l e d e l ' ~ l e c t r i c i t 6 , ov. 1 9 5 0 , p p. 4 79 -5 01 . 4 2 1 -4 2 6 .6 . H e t e n y i , M .; O n S i m i l a r i t i e s b e tw e e n S t r e s s a n d F l ow

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