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A study of fatigue stresses in marine propellor shafting Leith, Willliam Cumming 1949

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LIS *> ft? A STUDY OF FATIGUE STRESSES IN MARINE PROPELLOR SHAFTING by WILLIAM CUMMING LEITH A Thesis Submitted In P a r t i a l Fulfilment of The Requirements For The Degree Of MASTER OF APPLIED SCIENCE In The Department Of MECHANICAL ENGINEERING Thesis Supervisor: Head of Department: THE UNIVERSITY OF BRITISH COLUMBIA October, 1949 A STUDY OF FATIGUE STRESSES IN MARINE PROPELLOR SHAFTING ABSTRACT This paper describes an invest i g a t i o n carried out to study the fatigue f a i l u r e s of a "keyed tapered-shaft assembly" as affected by the keyway. Two fatigue t e s t i n g machines were b u i l t and used: one tested a keyed assembly i n reversed bending, and the other tested a keyed assembly i n reversed torsion or a combination of reversed bending and reversed tor s i o n . Both sled-runner and round-ended keyways were tested and the f a i l u r e s were compared with a view to establishing a law of f a i l u r e . ACKNOWLEDGEMENT The write r wishes to thank Professor W.O. Richmond fo r h i s guidance and. advice during t h i s i n vestigation; Professor W. Opeohowski f o r his help i n obtaining recent papers on the atomic theory; The B r i t i s h Columbia E l e o t r i o Railway Company Limited f o r the scholarship under which the research was conducted; and Mr. J.D. L e i t h f o r h i s assistance i n constructing the two te s t i n g machines. TABLE OF CONTENTS Page 1. Introduction 1 2. Theories of Strength 3 (a) Atomic Theory 5 (b) Maximum Shear Theory 7 ( i ) Assumptions 7 ( i i ) P r i n c i p a l Stress Equations ... 7 ( i i i ) Assumed law of F a i l u r e 9 3. Analysis of a Keyed Propellor Assembly 14 (a) Theoretical Method 14 (b) Empirical Method 16 (o) Photoelastio Method 19 (d) Modifications of "Liberty" Keyway .... 22 4. Fatigue Testing Machines 24 (a) Reversed Bending Machine 24 (b) Reversed Torsion Machine 27 5. Observations 37 6. Results 38 7. Conclusions 43 8. Recommendations 44 9. Appendix A 45 10. Appendix B 47 11. Bibliography 50 LIST OF ILLUSTRATIONS Page F i g . 1. A Typical T a i l s h a f t F a i l u r e 2 2. Types of Keyways 4 3. Normal and Shear Stresses 8 4. Mohr's Sphere 8 5. Mohr's Stress Plane 8 6. A law of F a i l u r e 10 7. Combined Stress Planes 10 8. Keyed Propellor Assembly of a "Liberty" Ship .. 15 9. Hydrodynamioal Analogy 16 10. Graph of Soap F i l m Method Results 17 11. Graph of Photoelastic Method Results 18 12. Keyed Shaft Assembly Model 20 13. Photoelastic Apparatus at U.B.C 20 14. Stress Patterns i n a Keyed Shaft Assembly 21 15. Original Keyway 23 16. Modified "Liberty" Keyway 23 17. Reversed Bending Machine 25 18. Reversed Bending Machine - General Arrangement. 25 19. Reversed Bending Machine - Deta i l s 26 20. Reversed'Torsion Machine - General Arrangement. 29 21. Torsion Arm and Eccentric 29 22. Reversed Torsion Machine - Details 30 23. SR-4 S t r a i n Indicator 31 24. Brush Oscillograph 31 (Cont'd) LIST OF ILLUSTRATIONS Page F i g . 25. Location of Strain Gages 32 26. Graphs of Cyclic Stresses 36 27. Reversed Bending Fractures 39 28. Reversed Torsion Fractures 40 29. Combined Reversed Bending and Torsion Fractures 41 30. Reversed Torsion Fracture at a F i l l e t 42 31. Caibration Curve for Torsion Arm 46 TABLES Page 1. Reversed Bending Data 28 2. Reversed Torsion Data 34 3. Combined Reversed Bending and Torsion Data . . . . 35 A STUDY OF FATIGUE STRESSES IN MARINE PROPELLOR SHAFTING The abnormal increase i n the number of f a i l u r e s occurring i n the t a i l s h a f t s of E.C.2-S-0.1 Liberty Ships b u i l t during the recent war, has caused much concern i n shipping c i r c l e s , 'in 1947 "Liberty 1* Ships (1)* represented about 20 per cent of the world gross tonnage or 13& per cent i n number. Information at 1st December, 1948, revealed that altogether a t o t a l of 583 "Liberty" screwshafts have been renewed, including during the past three years about 100 casualties at sea with r e s u l t i n g loss of propellor. The oost of these breakdowns i n i salvage and demurrage charges alone needs no emphasis. Most of the f a i l u r e s which have occurred near the large end of the t a i l shaft cone as shown i n F i g . 1,seemed to be of two types; one due to oorrosion fatigue and the other due to v i b r a t i o n stresses. Corrosion fatigue i s usually indicated by a circumferential groove around the shaft at the end of the i - See Bibliography | F i g . 1 A Typical Tailshaft Failure bronze l iner and i s caused by a defective sealing r ing . By 1947, failures from this cause had prac t ica l ly been eliminated. When the propellor has been slack on the taper, a bronze deposit i s usually found on the shaft indicating rubbing between the surfaces. Exoessive bearing on the sides at the forward end of the keyway usually formed cracks at the root of the keyway. Vibration stresses both torsional and bending, w i l l increase beyond safe l imi t s during excessive raoing of the propellor. These stresses acting on the existing stress raisers can weaken the shaft by overstress and promote rapid fa i lure by early cracking. Therefore, one can understand why fatigue cracks start at the forward corners of the keyway and on each side of the tapped hole for the forward key - retaining bol t . • 3 Two fatigue t e s t i n g machines were b u i l t with a view of studying the strength of keyed j o i n t s i n shafts; one tested a keyed tapered-shaft assembly i n reversed bending, and the other tested a keyed assembly i n reversed t o r s i o n or a combina-t i o n of reversed bending and reversed t o r s i o n . Tapered-shafts with sled-runner and round-ended keyways as shown i n F i g . 2 were tested and a study was made of the types of fractures obtained by the d i f f e r e n t methods of stressing. THEORIES OF STRENGTH Since t h i s paper describes an investi g a t i o n of the fatigue f a i l u r e of keyed connections, a theory of strength i s given to provide the academic background. For many years engineers and ph y s i c i s t s have attempted to correlate the atomic bond strength and the y i e l d strength of s t e e l . As yet, no one has formulated a theory to explain i t s a t i s f a c t o r i l y . However, X-ray d i f f r a c t i o n i s being used to study the effect of stress on the c r y s t a l struoture. Most of the values of strength used today are based on the experimental r e s u l t s of t e n s i l e , bending, impact, and t o r s i o n t e s t s . A ) R O U N D - E N D E D k f c r WAY V B) S I X P - R U N N E R . K X W A Y FlGj. 2 "[yP£5 OF KjE*YWAY5 5 ATOMIC THEORY OF STRENGTH Although none of the present-day atomic theories explain fatigue i n s t e e l , the Dislocation Theory (2) does agree with experimental r e s u l t s f o r s l i p , work hardening, and yi e l d i n g . 'Elementary t h e o r e t i c a l considerations, then, give us the following s t a r t i n g points f o r an atomic theory of strength (3). (a) a perfect c r y s t a l with no d i s l o c a t i o n , shouibd be very strong; s l i p would not take place u n t i l a shear s t r a i n of several degrees had been applied. (b) an otherwise perfect c r y s t a l containing a few d i s l o c a -tions would be very weak, since d i s l o c a t i o n s may be shown to move under the influenoe of a small shear stress. The strength of metals, and the rate of flow and oreep w i l l be determined by two factors ( i ) the rate at which disl o c a t i o n s are formed (ii) the resistance to t h e i r motion In discussing ( i ) i t must be remembered that a di s l o c a t i o n i s a long l i n e of m i s f i t , extending r i g h t through a c r y s t a l . I f the cohesive forces binding two c r y s t a l s together are nearly as great as those between planes of atoms i n the body of the c r y s t a l , the s t r a i n energy along an edge of t h i s type w i l l be nearly as great as that of a d i s l o c a t i o n , and l i t t l e energy w i l l be required to form a d i s l o c a t i o n there and 6 to move i t away. I f th is view i s correct, crystal boundaries are the sources of dislocations; i n single crystals , boundaries between elements of the mosaic must f u l f i l the same ro le . The condition for the formation of a dislocation i s that cohesion across the crystal boundary shal l give as much energy as cohesion within the c rys ta l , and this w i l l be the case only i f a l l the atoms are i n equivalent positions. In discussing ( i i ) , the factor preventing the motion of dislocations i s the presence i n the crystal of internal strains. Taylor suggested that the condition for the movement of a dislocation along a guide plane i s that the relevant component of shear s t ra in shal l actually have the same sign at a l l points along the guide plane; i f i t does not, the dis loca-t ion w i l l come to rest at a posit ion of equilibrium. Quali tat ively, th is theory leads to the conclusion that the y ie ld strength of a metal i s of the order of magnitude where G\ i s the elast ic shear modulus and «i i s the mean internal s t ra in . *A fatigue fa i lure (4) involves three stages (a) s l i p occurs and results i n s t ra in hardening and l a t t i ce distortions (b) the.fatigue crack starts (c) the crack spreads along the path of least resistance, due to stress concentration. This proceeds untiib the cross section of the metal i s reduced so much that the remaining portion breaks under the s ta t ic load. Hence fatigue fai lures usually exhibit two zones; the b r i t t l e 7 zone due to the r e a l fatigue action, and the fibrous zone having the same appearanoe as a f a i l u r e under a s t a t i c stress. MAXIMUM SHEAR THEORY OF STRENGTH Although we know that s t e e l i s not an i d e a l material, we apply the mathematical theory of e l a s t i c i t y ( 5 ) i n general design. I t s formulae are based on the following assumptions: (a) s t e e l i s homogeneous or i t can be divided i n f i n i t e l y i n t o smaller p a r t i c l e s without changing the strength s t i f f n e s s properties. (b) s t e e l i s i s o t r o p i c or i t has equal e l a s t i c s t i f f n e s s i n a l l d i r e c t i o n s . (c) s t e e l follows Hooke's law. In the most general case, the stress conditions of a element of a stressed body i s defined by the magnitude of the three p r i n c i p a l stresses , ts^ , o^ras shown i n F i g . 3. The algebraic values of the p r i n c i p a l stresses i s assumed to have the following r e l a t i o n , i n which the tension i s taken p o s i t i v e and compression negative. <^ x > > The maximum shear theory states that y i e l d i n g begins when the maximum shearing stress beoomes equal to the maximum shearing stress at the y i e l d point i n simple tension. Sinoe the maximum shearing stress i s equal to half the difference between the maximum and the minimum p r i n c i p a l stress, the 8 condition f o r y i e l d i n g i s IT* 6*$} - I** ^ ' f -The shearing stress (6) on a cross-section p a r a l l e l to the ^ plane may be divided into two components m andT*^ , p a r a l l e l respectively to the ^ and £ axes. In t h i s method of designation of stress components, the single subscript of a normal stress suoh as J*, and the f i r s t subscript of a shearing stress suoh as correspond with the d i r e c t i o n of the normal to the section, while the second subscript of ' T^ j indicates the d i r e c t i o n i n which the oomponent i s to be taken. The values of the p r i n c i p a l shearing stresses are defined as the r a d i i of Mohr's three p r i n c i p a l c i r c l e s shown i n F i g . 4 and 5 . For the general state of stress, the maxi-mum shearing stress occurs at sections making angles of 4 5 degrees with the sections across which the p r i n o i p a l stresses act. Timoshenko ( 5 ) desoribes a simple procedure to estimate the fatigue strength for d i f f e r e n t combinations of variable and steady loads. This method i s based on: (a) the maximum shear theory f o r d u c t i l e materials (b) the y i e l d stress f o r steady stresses (c) the endurance l i m i t f o r variable stresses (d) stress concentration i n d u e t i l e materials i s important fo r variable loads only. (e) the assumed law of f a i l u r e f o r oombined steady and variable stresses i s shown i n F i g . 6 and i s conserva-t i v e compared to (tough's experimental r e s u l t s . a z A S S U M E D " <5AFC U t M l f UlNE. ASSUMtO l_AW OF FMLUR.E t So • A L A W . o r FAILURE R< s . 7 COM&INED 5TR£SS PLANES II Se • endurance l i m i t f o r reversed stress ( f a i l u r e condition f o r complete load reversal) Sy - y i e l d point stress i n tension n • f a c t o r of safety The equation of the f a i l u r e l i n e i n terms of the steady stress So and the c y c l i c stress Sv i s : So _ J _ Sv s . Sy ^ Se" 1 I f we introduce n, the equation of the safe l i m i t l i n e i s : So Sv _ i Sy Se " 5; Applying these assumptions to shaft design, we s h a l l consider some important oases. Case 1 a steady torque To and a c y c l i c torque Tv loading = To- Tv s i n wt shear stress _ , ,„ _ (Ss)o = T o d . 16T0 due to To 2 J ' r r d shear stress (Ss)v =kTvjd s k i 6 Tv due to Tv 2J -77-</3 k • stress concentration factor due to a f i l l e t , keyway, etc. Since (Ss)o . (Ss)v = 1 (Ss)y ^ (Ss)e n I6T0 4 . K16Tv = ! 77"J3 (Ss)y VW 3 (Ss)e n I * (Ss)y * §Z (Ss)e = Se 2 2 Henoe d = (To + kTv)' ~ * rr (Sy Se) Case 11 a steady moment Mo and a c y c l i c moment Mv loading. = Mo"t Mv s i n wt normal stress So • Mod _ 52 Mo due to Mo 2 1 T T o / " 3 normal stress Sv = KMv cl a k52Mv due to Mv 2 1 Since So . Sv - 1 Sy Se n 52 Mo k32 Mv - 1 W 3 Sy ^ 3 Se n Henoe d = /»§& (Mo^kM^J / 7 T (Sy Se) Case 111 a steady torque To, a steady moment Mo, a o y c l i c torque Tv, and a c y c l i c moment Mv loading = (Mo-Mv s i n wt)+ (TotTv s i n wt) In t h i s case we have combined normal and shear stresses which are shown i n F i g . 7 f o r a given plane P. P. normal stress due . . , „, . Sx - M o & + kMv d to Mo^ Mv 2 1 2 1 shear stress due m , m „ Sxy a To d ± kTv d to To - Tv 2 J 2 J Ss = combined shear stress dm plane P. P. = AC s i n 2(0+-A) » Sx s i n 2A •+• Sxy cos 2A 2 Hence Ss = (Mo d ... kMv d) s i n 2A _ L . (Tod + kTvd) oos 2A ( 41 41 ) ( 2J 2J ) (Ss)o « Mo d s i n 2A To d cos 2A 41 ZZ (Ss)v • kMv d s i n 2A_i_ k Tv d cos 2A 41 2J Substituting 1 = n 1 = (Mod s i n 2A Tod oos 2A) (kMvd s i n 2A kTvd cos 2A) n ' 41 2J 41 - 2J (Ss)y (Ss)e Hence 3 & s/52n(Mo s i n To oos 2A kMv s i n 2A-+-kTv oos 2A) / 7T (— s y Se ) where (Ss)y = §L , (Ss)e = Se ; tan 2A = 1/2 (Mo+k Mv) 2 2 (To+k Tv) ANAYLSIS OF A KEYED PROPELLOR ASSEMBLY The fatigue strength of a keyed propellor assembly as shown i n P i g . 8 oan be estimated by combining the t o r s i o n a l , bending and thrust stresses f o r i d e a l conditions of loading, that i s , calm seas and a f u l l y submerged propellor. But there are a number of indeterminate stresses which may occur i n d i -v i d u a l l y or together; the maximum t o r s i o n a l v i b r a t i o n stresses when the propellor i s racing, the maximum bending stresses when the ship i s i n b a l l a s t , and the clamping stresses at the top of the taper by the edge of the propellor boss recess and by the edge of the shaft l i n e r . A complete t h e o r e t i c a l solution f o r the stress con-centration at a keyway does not e x i s t , but the "hydrodynamloal analogy" ( 5 ) i s useful f o r disoussing the to r s i o n of shafts. The twisting of shafts of uniform cross-section i s mathemati-c a l l y i d e n t i c a l to the motion of a f r i c t i o n l e s s f l u i d moving with uniform angular v e l o c i t y inside a s h e l l having the same section as the bar. The v e l o c i t y of the c i r c u l a t i n g f l u i d at any point i s taken as representing the shearing stress at that point of the cross section of the bar when twisted. For the case of a keyway with sharp corners (See Fig.9) t h i s analogy indioates a zero v e l o c i t y of the c i r c u l a t -ing f l u i d at the outside corners (points M-M); hence the shearing stress i n the corresponding t o r s i o n problem i s zero, at such points. S i m i l a r i l y , t h i s analogy indicates an i n f i n i t e v e l o c i t y at the inside corners (points N-N), the vertioes of I I i I I " U b E R T Y * S H I P the reentrant corners; henoe the shear-ing stress i s also i n f i n i t e at suoh points. Sinoe a keyway with absolutely sharp corners cannot be cut, the reentrant corners w i l l have a small F i g . 9 Hydrodynamioal radius of f i l l e t and a f i n i t e stress Analogy d i s t r i b u t i o n . A stress concentration factor WK" i s used to denote the r a t i o of the maximum stress to the nominal stress; that i s , K = m a x ' o-* nom. Figs. 10 and 11 show published values of the stress concentra-t i o n factor "K" as have been obtained by the soap f i l m method and the photoelastic method of measuring stress. Several experimenters have carried out endurance test s to determine the nominal stress concentration factors fo r sled-runner and round-ended keyways, but because of s i z e effect and clamping stress S. Archer (1) maintains that u n t i l large-scale tests of f u l l - s i z e propellors are made, there seems to be l i t t l e to choose between the two types of keyways. Under steady conditions i n smooth water, with some bending fatigue present, such as when the propellor i s p a r t l y submerged i n the ballasted conditions, i t i s estimated from the evidence ( l ) so f a r available that the fatigue strength of a w e l l f i t t e d propellor assembly, expressed i n terms of nominal reversed t o r s i o n a l stress, may be of the order of 6000 p s i . I t should always be borne i n mind, however, that t h i s estimat-ed value may be seriously reduced i n service on aooount of a variety o f oauses, such as extremes of stress concentration O.ID = h i GRIFFITH 8 TAYLOR I HOLLOW SHAFT (SOAP FILM METHOD) TECH. REPT. NACA. VOL.3,1917- 1918 -g- > .05 : MAXIMUM STRESS APPROACHES A. •jj < .05 : MAXIMUM STRESS AT B. K = ,WHERE N^OM. T N Q M =MAX. SHEAR STRESS CALCULATED FOR SHAFT WITHOUT KEYWAY. R q . l O < ^2.5 SOLID SHAFT: A.S.BOYD (SOAP FILM METHOD) CURVE 304511-Dj aris ing from machining erros, badly f i t t ed parts, corrosive envirnment, and previous overstress. Stress concentration at a keyway (7) i s best r e a l i z -ed with the aid of photoelasticity (8) which u t i l i z e s the opt ical effeot of stresses on a transparent model. Certain transparent materials (9) when stressed become birefringent, that i s i f a beam of l igh t be passed through the stressed material i t i s s p l i t up into two plane polarized waves of l ight which vibrate in planes at r ight angles to each other. At each point i n the material the two planes coincide with the direc-tions of the pr incipal stresses, and the birefringence persists only while the material i s stressed. The waves are each retard-ed i n passing through the material, the amount of retardation depending on the corresponding pr incipal stresses and on the thickness of the material. Thus, on emergence, the two waves are retarded relat ive to one another, or out of phase, by an amount proportional to the difference of the pr inc ipal stresses. To observe the relat ive retardation i t i s necessary to resolve the two component waves in one direct ion. This may convenient-l y be done by using an analysing unit i n the emergent beam of l i g h t . Due to the phase difference between the two components an interference phenomenon w i l l then be produced, and at each point where the phase difference i s half a wave length, there w i l l be complete extinction. Such points cause dark bands which w i l l be v i s ib l e i n the form of fringes on a suitable screen. Therefore by using a two dimensional model, the resul t-ing fringe pattern i s an indication of the pr inc ipal stress No Load Load - 2 lbs . Load - 4 lbs . Load - 5 lbs. F i g . 14 Stress Patterns in a Keyed Shaft assembly difference at a l l points i n the model. By suitable c a l i b r a t i o n the stress difference per fri n g e oan be determined. The writer made the keyed shaft assembly shown i n Fi g . 12 and studied i t i n the photoelastic apparatus of the Mechanical Department shown i n F i g . 15. The stress concentra-tions at the reentrant corners are especially obvious i n Fig.14 and demonstrate the detrimental effect of keyways. Some revisions i n the keyway design of "Liberty 1* t a i l s h a f t s have been made to increase t h e i r fatigue strength. The o r i g i n a l keyway as shown i n F i g . 15, a trac i n g of the Burrard Dry Dook sp e c i f i c a t i o n s of 1943, was of the round-ended type with no regulations governing the f i l l e t . In 1947, Mr. W. 0. Richmond (15) recommended a sled-runner keyway with a rounded f i l l e t ( t h i s was l a t e r adopted) and shot-peening of the forward end of the keyway. The revised keyway as shown i n F i g . 16, a t r a c i n g of the Baldwin Looomotive Works sp e c i f i c a t i o n s approved by the American Bureau of Shipping i n 1948, i s of the sled-runner type with a 5/16 i n . f i l l e t and has i t s edge broken with a f i l e . The key has a 1/4 i n . chamfer on a l l sides and i s to be held i n place by two jack-screws. In addition, the key has 1/8 i n . saw cuts at the forward corners to transfer the load more equally throughout i t s length. Furthermore, t a i l s h a f t s are to be drawn and inspected every two years rather than three. FATIGUE TESTING MACHINES 24 Economy and machining d i f f i c u l t i e s l i m i t e d the test bar to a 3/4 i n . diameter shaft with a 3/16 i n . keyway. Using t h i s shaft s i z e as a basis i n the design, the two t e s t i n g machines were b u i l t to subject keyed tapered-shafts to repeated reversed stresses. In a l l t e s t s , a known load was applied and the number of stress cycles was measured with a counter. Since these machines were constructed as preliminary models, a l i s t of suggested improvements i s given i n the Recommendations. The wr i t e r wishes to acknowledge that both machines were suggested by current l i t e r a t u r e : the bending machine i s s i m i l a r to one developed at the University of I l l i n o i s (10), and the eccentric of the torsion machine resembles one designed by Holley (11). REVERSED BENDING MACHINE The reversed bending machine i s e s s e n t i a l l y a piece of s t e e l tubing mounted between two b a l l bearings. One end i s driven by a motor and the other end contains a tough s t e e l bushing which supports the t e s t shaft. The r o t a t i n g test bar mounted as a cantilever beam, i s loaded transversely with a known weight as shown i n F i g s . 17 and 18. F i g . 19 shows the d e t a i l s of the machine. The large end of the tapered cone, which i s the o r i t i c a l section i n loading, has shearing stresses as well as F i g . 18 Reversed Bending Machine - General Arrangement 6 > ! L L O F M A T E 1 P J / M _ No. I T T U & E ! 4 Cb.uPu M<£. 7 8 ; *3i f . i B R A C K E T B R A C K E T " & O L " W.ASH.E.PL R r F E . R H . U C E L . B " £ ? ' H "I 15 — ift^r i. T I i i 1 en. -t; I REAMED H O L E S F O R . ITEM "3 i.2: 2- M- 5- B c A f c i M q CASES . ,MK. .ft •e-( i HP ^ j (' _ i . — i — /4 ''4-' 5 E T bCRJflVN'S 3 1" M ' 5 - COUPLING, H i C P p ^ S S F \ l S M T O M K - A §1 / ... •• t LJ_j. i i f f ± te 13 i f - | 4 . 1 - 5 T - S ' COLLAR. M O 8 -1 L 9 i -M-S- WASHER. M K I i r i > i t ^ 3/ " "4 O ' D - ' S T ^ - ^ R*&»**8 5 % * 1; i " M:S-. TU B E . HK . . A M A C H I N E To Frr ^ " K . D " 2 _ 33. i d -2' (5 1- Tt S T O A R H O L E I N M K . F ~ 3 L O T I N MicGj I N \ \ \ A. HI i—•r 3 artta i f 11 .8 ; - M - S - BOLT M K . H ' MECHANICAL ENQINEE.R.1.NG UNIVERSITY OF &R.ITISH COLUMBIA '6 j 1 - M - S - BRACKET- M t c F (7 1 - M - S - BRACKET MKJ'G REVER.SED.BENDING MACHINE S C A L E . . D R A W H 6X U T " C MAY 1 ^ 4 ^ DRAWING NO. 7-1 the maximum bending stresses acting on i t . However, the stress concentration at the forward end of the keyway usually causes f a i l u r e there. The nominal stresses were calculated f o r each specimen and are l i s t e d i n Table I. As the maohine ran smoothly for a l l t e s t s , no modifi-cations were necessary. REVERSED TORSION MACHINE The reversed t o r s i o n maohine consists mainly of the torsion arm which twists one end of the test bar, while the other end i s held f i r m l y i n the tor s i o n block. A l i n k connects the arm to the eccentric plate. By adjusting the slotted eccentric plate, the angular displacement of the arm and hence the applied torque, oan be varied. The eccentric i s mounted on a shaft d i r e c t l y connected to a 3/4 hp Baldor variable-speed motor; for d e t a i l s of t h i s motor ref e r to (12). F i g . 20 shows the general arrangement of the maohine and F i g . 21 shows a view of the eccentric. The d e t a i l s of the maohine are given i n Fi g . 22. The maohine was fastened to the concrete f l o o r with anohor-bolts to overcome the v i b r a t i o n caused by the unbalance of the e c c e n t r i c Also, the running speed was reduced from 1800 to 1000 rpm to lessen wear at the li n k - e c c e n t r i c connec-t i o n . However, a l i t t l e movement which was noticed i n the torsion block was assumed due to f l e x i b i l i t y i n the maohine i t s e l f . 28 TABLE 1 REVERSED BENDING DATA Number Keyway Bending Stress Shear Stress Number of Cycles Remarks 1 ' SR 13600 113 22624000 UNBR « 21700 171 507000 BR 2 • SR 16300 128 5012000 BR ... .3 ' SR 14000 16800 131 156 iq0417000_ 2204000 UNBR BR 4 RE 16200 128 1044000 BR 5 RE 13600 128 3430000 BR ; . 6 RE 10800 _ 13600 85 106 30577000 j UNBR , 38180~00 T BR | SR - sled-runner keyway RE m round-ended keyway BR s broken * UNBR B unbroken Bending Stress r MP. Shear Stress A AB I 9 10 PLAN =3 ft 3 L t . ' 7 4 -E L ELVATION P^. S L O T ^4-' R E A M to 2£ L 8 m E N D V i e w 4- KflYWAf i 1 1 - M -5 • E O E K T W C PLATE: A (% 1.-M-5- EC C E N T W C Hub M*Jd a 1 - 10 DE T A I L O F C C C E . N T R 4 C E J ®5 74-i b l L L O F M A T E 1 W A L . V% + REAMED I S ' ® l - M ' 5 - TORS\OM Pm^ H^F « 12' • 7 % S ^ U R S C J E : S H O U L D E R . iff! KG II \\ 7 _ 1 L ± 1 5 - T E S T & A * . M K . Q . No. n U -P A R T QUAKL REFE.^£:NCE. i ECC&NTWC PLATE 1 , MK. "A° ; Z . far&qmt hue 1 3 "TOBSXOTH B L O C K . ; 1 i& •£>E1AR-\M^. 2. " 0 " BEARJNG, " E . , L s ."foPSvON ARM i " p u 8 I 9 3 / 4 - ff3 M O T O ^ 1 IO xi8" S H A F T 1 J J _[ '/a> bouT f *• 12 • % DP-ILL-4" 3 I ' M ' 5 ' JORS\ON ftLQOc M K J ' C §T §s—; o V ^ 1 u 4 1 -H-S- 5 E A R t N ^ - ' H l J D ' i 1 -H-S- B&ARJNQ MK."E" ^ ' REAMED - » 9 6 ' 8 l - H - 5 - LINK. M K . H MECHANICAL ENGINEERING UNIVERSITY O F ftHiTiSH COLUMBIA D R A W N S>Y~~ O T " C. aSxtDJ D R A W I N G ^40. FlQ.22 HlllHimi^-^H F i g . 23 SR-4 Strain Indicator 34 — F i g . 24 Brush Oscillograph 3Z The strain and hence the stress in each specimen was calculated from the applied torque which was measured by SR-4 Strain Gages on the torsion arm. These Gages are made of resistance-strain-sensitive wire bonded to a thin paper carrier they were cemented on the edges of the torsion arm and covered with wax as shown in F i g . 21. When the arm was displaced, one of the Gages was strained in tension and the other in compres-sion. The change in resistance which i s proportional to the change in s t rain, can be measured and converted to stress. Therefore, knowing the strain in the torsion arm, the stress in a specimen could be calculated from the cal ibrat ion curve shown in Appendix A. The stat ic strains (when the machine was stopped) were measured with the SR-4 Strain Indicator shown in F i g . 23. The dynamic strains (when the machine was running) were measured with the Brush Oscillograph as shown in F i g . 24. F i g . 25 Location of Strain Gages 33 Since the s t a t i c and dynamic stresses agreed w i t h i n experiment-a l error, the oscillograph curves were used to measure the applied torque. Also, these nominal stresses agreed with the actual stresses i n the test shaft measured with a separate s t r a i n gage shown i n F i g . 25. A complete description of the measuring instruments i s given i n a paper by Johnson and Bruce (12). The nominal stresses were calculated for each specimen and are shown i n Table I I . Combined reversed bending and reversed t o r s i o n a l stresses were imposed on a test shaft when the two bearings supporting the to r s i o n shaft were removed. The t o r s i o n a l stresses were calculated from the applied torque and the bend-ing stresses were measured by s t r a i n gages on the t e s t shaft. The nominal stresses were calculated f o r each specimen and are shown i n Table I I I . Graphs of the oyolic stresses as measured by the Brush Oscillograph are shown i n F i g . 26. TABLE 11 REVERSED TORSION DATA Number Keyway Torsion Stress Number of Cycles Remarks 1 RE 8500 1022E00 UMBR 9400 261700 BR 2 SR 9400 230000 BR at f l a t s 4 SR 8400 275000 BR at f i l l e t 5 RE 8000 312000 BR 6 SR 8000 1163500 BR j » SR s sled-runner keyway RE s round-ended keyway BR r broken UNBR = unbroken Torsion Stress = ~ 3S-TABLE 111 COMBINED REVERSED BENDING AMD TORSION DATA Number Keyway Bending Stress Torsion Stress Number of Cycles Remarks 3 SR 60000 4000 8700 BR 7 RE 3300 9400 . 14100 BR 8 SR 3300 9400 41000 BR SR a sled-runner keyway RE - round-ended keyway BR m broken UNBR = unbroken Bending Stress = Torsion Stress = ~ 3 6 Torsion Test Torsion Stress Combined Bending and Torsion Test Torsion Stress Combined Bending and Torsion Test Bending Stress F i g . 26 GRAPHS OF CYCLIC STRESSES 37 OBSERVATIONS 1. At f i r s t the bearings supporting the eccentric shaft heated up and threatened to seize when the maximum e c c e n t r i c i t y was applied. On the advice of Professor Wm. Wolfe, Sulfur Flowers was added to the o i l . Immediately the bearings cooled to a moderate temperature and the wear from then on seemed n e g l i g i b l e . Therefore, Sulfur acts very w e l l as an extreme pressure lubricant, f or s t e e l shafts i n bronze bushings. 2. A l l of the shafts tested i n the tor s i o n machine, especially ones with sled-runner keyways heated up to a temperature that could hardly be touched. Since t h i s i s due to the work done, i t may have some effeot on the fatigue strength as the shafts were hotter f o r higher applied stresses. RESULTS 1. In a l l oases the sled-runner keyway withstood more cycles at a given stress than the round-ended keyway. 2. Reversed Bending Fractures are shown i n F i g . 27. (a) the sled-runner keyway showed a circumferential orack on the d r i v i n g side near the forward end. lb) the round-ended keyways broke near the middle of the taper probably r e s u l t i n g from poor f i t s and a series of power f a i l u r e s . 3. Reversed Torsion Fractures are shown i n F i g . 28. (a) the sled-runner keyway had a crack s t a r t at the root of the forward corners and progressed at 45 degrees to the shaft a x i s . These cracks became v i s i b l e at about 500,000 cycles and the writer watched them progress u n t i l f a i l u r e at 1,163,500 oyoles. This specimen f a i l e d exactly as predicted by the maximum shear theory. (b) the round-ended keyway had a r a d i a l crack s t a r t at each side of the forward end at approximately 45 degrees to the shaft a x i s . 4. Combined Bending and Torsion fractures are shown i n F i g . 29. Both types of keyways had cracks at the forward end and showed some evidence that the angle of fracture i s governed by the 3«? F i g . 26 Reversed Bending Fractures F i g . £7 Reversed Torsion Fractures F i g . 28 Combined Reversed Bending and Torsion Fractures combined stresses. 5. The reversed torsion fracture at a f i l l e t as shown i n F i g . 30 had a circumferential crack at the stress r a i s e r with r a d i a l cracks pointing towards the centre. •Fig. 30 Reversed Torsion Fracture at a F i l l e t 6. Sulfur was e f f e c t i v e l y used as an extreme pressure l u b r i -cant for s t e e l on bronze. C O N C H T S I O H S 43 1. A sled-runner keyway has a higher fatigue strength than a round-ended keyway i n a keyed tapered-shaft assembly stressed by reversed bending, reversed torsi o n , or combined reversed bending and reversed t o r s i o n . 2. The maximum shear theory of strength predicts the plane of fatigue f a i l u r e . (a) bending stresses produce a circumferential fracture perpindicular to the shaft a x i s . (b) t o r s i o n a l stresses produoe a h e l i c o i d a l fraoture at 4 5 degrees to the shaft a x i s . (o) combined bending and t o r s i o n a l stresses produce a fraoture whose angle of i n c l i n a t i o n to the shaft axis r i s governed by the combined stress. 3. Owing to s i z e effeot, the difference i n fatigue strength between the two types of keyways may be less prominent i n the actual shaft. 4 . The measuring instruments showed that the s t a t i c and dyn-amic stresses are equal i n the torsion machine. 5 . Since l i t t l e research has been done on the strength of keyed connections, these machines could be useful f o r further investigations on t h i s subject. RECOMMENDATIONS 1. R o l l e r bearings would be suitable f o r the two bearings supporting the eccentric shaft rather than the bronze sleeve. 2. I f the to r s i o n arm was reduced to 10 i n . of e f f e c t i v e length, higher speeds and higher stresses could be used; or fo r the same conditions, smoother operation would r e s u l t . 3. For an exact determination of the fatigue strength of keyed connections, the shaft and the bushing should be ground to s i z e . Vancouver Engineering Works has the equipment to do t h i s job. 4. By the use of suitable bushings, both machines oould be modified to test straight keyed connections or press f i t s . APPENDIX A 45 The SR-4 S t r a i n Gages which measured the applied torque on the torsion arm were calibrated by s t a t i c loads using the Baldwin SR-4 S t r a i n Indicator. This indicator con-, s i s t s of a four arm Wheatstone bridge. The s t r a i n gages are connected so that the bridge c i r o u i t i s unbalanced when a change i n stress changes the gage resistances. When a gage i s strained i n tension i t s length increas-es and i t s diameter decreases with a resultant change i n e l e c t r i c resistance. The r a t i o between the "change i n r e s i s t -ance" to the "change i n s t r a i n " i s c a l l e d the s t r a i n s e n s i t i v i t y or the gage factor F. The unit s t r a i n i s equal to the unit resistance change divided by the gage fact o r . This indicator reads the s t r a i n d i r e c t l y i n micro inches. F i g . 31 shows the c a l i b r a t i o n ourve f o r the torsion arm to be l i n e a r as expected. CALIBRAT ST ION OF TOR RAIN GAGES 3I0N ARM 46 CQ <D w> <s F i g . 31 •sion Arm ( 1 - . C > C > C >nce On Toi licroinche! ) c / ior Differ* r > c - -Indioa-> c <3UU ul rt 100 -0 30 00 60 Shear 00 90 Stress i n p s i . DO 12 rest Shaft D00 15 300 APPENDIX B 47 To show that the s t a t i c and dynamic stresses i n a torsion t e s t shaft agreed within experimental error. The s t a t i c strains were measured with the SR-4 S t r a i n Indicator. The dynamic st r a i n s were measured with the Brush Oscillograph. Nominal Stresses The nominal stresses were calculated f o r the applied torque measured by the s t r a i n gages on the t o r s i o n arm as shown i n F i g . 25. la) S t a t i c Load Maximum Reading = 9920 micro-inches Minimum Reading • 8920 " Difference - 1000 H e B s t r a i n r 500 micro-inches From the calibration curve i n Appendix A SI - ± 13600 p s i . (b) Dynamic Load E-t- AE V - 6 v o l t s From i n i t i a l setting 15 mm de f l e c t i o n = 5 m i l l i - v o l t s 7 mm " 2.33 " 48 AR - 2AE a 8(.00235) R V 6 10078 ohms per ohm e 1 AR F R 2.06 1_ (.0078) _ .000380 inches F s gage factor n 2.06 from c a l i b r a t i o n curve i n Appendix A S, = - 11000 p s i . Measured Stresses The measured stresses were calculated from the s t r a i n observed i n a s t r a i n gage at 45 degrees to shaft axis as shown i n F i g . 25. (a) S t a t i c Load Maximum Reading - 1985 micro-inches Minimum Reading s 815 " Difference = 1170 » e 585 micro-inches e i13500 p s i (b) Dynamic Load v 1 E = i E+A E = AR -V = (R+«R) A E = AE = V IR-HAR) V 2R+AR 2 V 4 AR R V 2 small 6 volts From i n i t i a l setting 15 mm deflection = 5 m i l l i - v o l t s 5.5 mm " • 1.83 H AR = 4 AE = 4 (.00183) = O Q . —y— J g— u .00122 ohms per ohm e - 1 AR = 1 (.00122) = F ~R 2706 " • 0 0 0 5 9 0 inches S, = I_JL_ = (30 x 10 6) (590 x 10"6) s + ' ~ 1+ m 1.3 -13600 ps i The wire from the s t ra in gages on the torsion arm had to be ©hanged twice because several of the strands broke from the vibration. A broken strand was usually indicated by an unsteady balance on the indicator and erratic curves on the oscillograph. BIBLIOGRAPHY SO 1. Archer, S., Screwshaft Casualties - The Influence of Torsional Vibration and Propellor Tmmersion, London, The I n s t i t u t e of Naval Architects and Marine Engin-eers, A p r i l 8, 1949. 2. S e i t z , Fredrick, The Physios of Metals, New York, McGraw H i l l Book Co., 1943. 3. Mott, N. F., Atomic Physios and the Strength of Metals. London, The I n s t i t u t e of Metals, Vol.72, Part 6, 1946. 4. Boas, W., Physios of Metals and A l l o y s . New York, John Wiley and Sons, 1947. 5. Timoshenko, S., Strength of Materials. New York, D. Van Nostrand Co., 2 vols, 1947. 6. Nadai, A., P l a s t i c i t y . New York, McGraw H i l l Book Co., 1931. 7. Solakin, and K a r e l i t z , "Photoelastio Study of Shearing Stresses i n Keys and Keyways". ASME, 1931. 8. Frocht, M. M., Photoelastioity, New York, John Wiley & Sons, 2 vols, 1941. 5/ 9. Shaw, F. S., "Determination of Stress Concentration Factors," A Symposium of the Fatigue of Metals. Melbourne, University of Melbourne, 1947. 10. Moore, H. F., and Krouse, G. N., Repeated-Stress (Fatigue) Testing Machines Used i n the Materials Testing  Laboratory of the University of I l l i n o i s . Urbana, 111. Exp. Sta., C i r c u l a r 23, 1934. 11. Holley, E. G., The S t a t i c and Fatigue Torsional Strengths of Various Steels With C i r c u l a r . Square, and  Rectangular Sections. London, The I n s t i t u t e of Mechanical Engineers, September 1940. 12. Johnson, W. J . , and Bruce, H. C , C y c l i c Stresses i n Marine Propellor Shafting. Vancouver, B. C , 1949. 13. Richmond, W. 0., F a i l u r e of T a i l Shafts of Victory Type Steam Cargo Ships. Vancouver, B. C , May 23, 1947. 14. Peterson, R. E., F a i l u r e of Shafts Having Keyways.ASTM. 1932. 15. Moore, H. F., The E f f e c t of Keyways on the Strength of Shafts 111. Exp. Sta., Urbana, B u l l e t i n 42. 16. Dorey, S. F.. Limits of Torsional Stress i n Marine O i l Engine Shafting. London, The Engineer, A p r i l 11,1947. 5Z 17. Gough, H. J . , The Fatigue of Metals. New York, D. Van MoBtrand Oo. Ltd., 1926. 18. B a t t e l l e Memorial I n s t i t u t e , Prevention of Fatigue i n Metals. New York, John Wiley & Sons, 1946. 19. Richmond, W. 0., Progress Report on Research Project -Cy c l i c Stresses i n Ship Propellor Shafting, Vancouver, B. C , February 12, 1949 

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