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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 I n P a r t i a l F u l f i l m e n t of The Requirements For The Degree Of MASTER OF APPLIED SCIENCE I n 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 i n v e s t i g a t i o n c a r r i e d out to study the f a t i g u e f a i l u r e s of a "keyed tapered-shaft assembly" as a f f e c t e d by the keyway.  Two f a t i g u e t e s t i n g  machines were b u i l t and used: one t e s t e d a keyed assembly i n reversed bending, and the other t e s t e d a keyed assembly i n reversed t o r s i o n or a combination of reversed bending and reversed t o r s i o n .  Both sled-runner and round-ended keyways  were t e s t e d and the f a i l u r e s were compared w i t h a view t o e s t a b l i s h i n g a law of f a i l u r e .  ACKNOWLEDGEMENT  The w r i t e r wishes t o thank P r o f e s s o r W.O. Richmond f o r h i s guidance and. advice during t h i s i n v e s t i g a t i o n ; Professor W. Opeohowski f o r h i s 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 L i m i t e d f o r the s c h o l a r s h i p under which the research was conducted; and Mr. J.D. L e i t h f o r h i s a s s i s t a n c e i n c o n s t r u c t i n g the two t e s t i n g machines.  TABLE OF CONTENTS Page 1  1. I n t r o d u c t i o n 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 S t r e s s Equations ...  7  ( i i i ) Assumed law of F a i l u r e  9  3. A n a l y s i s of a Keyed P r o p e l l o r Assembly  14  (a) T h e o r e t i c a l Method  14  (b) E m p i r i c a l Method  16  (o) P h o t o e l a s t i o Method  19  (d) M o d i f i c a t i o n s of " L i b e r t y " Keyway ....  22  4. Fatigue Testing Machines  24  (a) Reversed Bending Machine  24  (b) Reversed Torsion Machine  27  5. Observations  37  6. R e s u l t s  38  7. Conclusions  43  8. Recommendations  44  9. Appendix A  45  10. Appendix B  47  11. B i b l i o g r a p h y  50  LIST OF ILLUSTRATIONS Page F i g . 1. A T y p i c a l 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 S t r e s s Plane 6. A law of F a i l u r e  8 10  7. Combined S t r e s s Planes  10  8. Keyed P r o p e l l o r Assembly of a " L i b e r t y " Ship ..  15  9. Hydrodynamioal Analogy  16  10. Graph of Soap F i l m Method Results  17  11. Graph of P h o t o e l a s t i c Method Results  18  12. Keyed Shaft Assembly Model  20  13. P h o t o e l a s t i c Apparatus a t U.B.C  20  14. S t r e s s Patterns i n a Keyed Shaft Assembly  21  15. O r i g i n a l Keyway  23  16. Modified " L i b e r t y " Keyway  23  17. Reversed Bending Machine  25  18. Reversed Bending Machine - General Arrangement.  25  19. Reversed Bending Machine - D e t a i l s  26  20. Reversed'Torsion Machine - General Arrangement.  29  21. Torsion Arm and E c c e n t r i c  29  22. Reversed Torsion Machine - D e t a i l s  30  23. SR-4 S t r a i n I n d i c a t o r  31  24. Brush O s c i l l o g r a p h  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 L i b e r t y Ships b u i l t during the recent war, has caused much concern i n s h i p p i n g c i r c l e s , ' i n 1947 "Liberty * Ships ( 1 ) * represented about 20 per 1  cent of the world gross tonnage or 13& per cent i n number. Information a t 1 s t December, 1948, revealed t h a t a l t o g e t h e r a t o t a l of 583 " L i b e r t y " screwshafts have been renewed, i n c l u d i n g during the past three years about 100 c a s u a l t i e s a t sea w i t h r e s u l t i n g l o s s of p r o p e l l o r .  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 l a r g e end of the t a i l s h a f t cone as shown i n F i g . 1,seemed t o be of two types; one due t o o o r r o s i o n f a t i g u e and the other due to v i b r a t i o n s t r e s s e s . C o r r o s i o n f a t i g u e i s u s u a l l y i n d i c a t e d by a c i r c u m f e r e n t i a l groove around the s h a f t a t the end of the i - See B i b l i o g r a p h y  |  F i g . 1 A Typical Tailshaft Failure bronze l i n e r and i s caused by a defective sealing r i n g .  By  1947, failures from this cause had p r a c t i c a l l y 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 i m i 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 f a i l u r e 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 b o l t .  •  3  Two f a t i g u e t e s t i n g machines were b u i l t w i t h a view of studying the s t r e n g t h of keyed j o i n t s i n s h a f t s ; one t e s t e d a keyed tapered-shaft assembly i n reversed bending, and the other t e s t e d a keyed assembly i n reversed t o r s i o n or a combinat i o n of reversed bending and reversed t o r s i o n .  Tapered-shafts  w i t h sled-runner and round-ended keyways as shown i n F i g . 2 were t e s t e d and a study was made of the types of f r a c t u r e s obtained by the d i f f e r e n t methods of s t r e s s i n g .  THEORIES OF STRENGTH  Since t h i s paper d e s c r i b e s an i n v e s t i g a t i o n of the f a t i g u e f a i l u r e of keyed connections, a theory of strength i s given t o provide the academic background. For many years engineers and p h y s i c i s t s have attempted t o c o r r e l a t e the atomic bond s t r e n g t h and the y i e l d strength of s t e e l .  As yet, no one has formulated a theory t o  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 t o study the e f f e c t of s t r e s s on the c r y s t a l s t r u o t u r e . 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 tests.  A ) R O U N D - E N D E D k f c r WAY  V  B)  S I X P - RUNNER.  FlGj. 2  "[yP£5 OF  K X W A Y  KjE*YWAY5  5  ATOMIC THEORY OF STRENGTH Although none of t h e present-day atomic t h e o r i e s e x p l a i n f a t i g u e i n s t e e l , the D i s l o c a t i o n Theory (2) does agree w i t h experimental r e s u l t s f o r s l i p , work hardening, and yielding. 'Elementary t h e o r e t i c a l c o n s i d e r a t i o n s , then, give us the f o l l o w i n g s t a r t i n g p o i n t s f o r an atomic theory of strength (3). (a) a p e r f e c t c r y s t a l w i t h 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 a p p l i e d . (b) an otherwise p e r f e c t c r y s t a l containing a few d i s l o c a t i o n s would be very weak, since d i s l o c a t i o n s may be shown t o move under the influenoe of a small shear s t r e s s . The strength of metals, and the r a t e of flow and oreep w i l l be determined by two f a c t o r s ( i ) the r a t e a t which d i s l o c a t i o n s a r e formed (ii) the r e s i s t a n c e t o t h e i r motion I n d i s c u s s i n g ( i ) i t must be remembered that a d i 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 crystal.  I f the cohesive f o r c e s 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 n e a r l y 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 t o form a d i s l o c a t i o n there and  to move i t away.  6  I f t h i s view i s correct, c r y s t a l boundaries  are the sources of dislocations; i n single c r y s t a l s , boundaries between elements of the mosaic must f u l f i l the same r o l e .  The  condition for the formation of a d i s l o c a t i o n i s that cohesion across the c r y s t a l boundary s h a l l give as much energy as cohesion within the c r y s t a l , and t h i s 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 c r y s t a l of internal strains.  Taylor suggested that the condition for the movement  of a d i s l o c a t i o n along a guide plane i s that the relevant component of shear s t r a i n s h a l l actually have the same sign at a l l points along the guide plane; i f i t does not, the d i s l o c a t i o n w i l l come to rest at a position of equilibrium. Qualitatively, t h i s theory leads to the conclusion that the y i e l d strength of a metal i s of the order of magnitude where G\ i s the e l a s t i c shear modulus and «i i s the mean internal s t r a i n . *A fatigue f a i l u r e (4) involves three stages (a) s l i p occurs and results i n s t r a i n hardening and l a t t i c e 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 t a t i c load.  Hence  fatigue f a i l u r e s usually exhibit two zones; the b r i t t l e  zone due t o the r e a l f a t i g u e a c t i o n , and the f i b r o u s zone  7  having the same appearanoe as a f a i l u r e under a s t a t i c s t r e s s .  MAXIMUM SHEAR THEORY OF STRENGTH Although we know that s t e e l i s not an i d e a l m a t e r i a l , 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 f o l l o w i n g assumptions:  (a) s t e e l i s homogeneous or i t can be d i v i d e d 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 s t r e n g t h stiffness 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 in a l l directions. (c) s t e e l f o l l o w s Hooke's law. I n the most general case, the s t r e s s c o n d i t i o n s 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 s t r e s s e s  , ts^ , o^ras shown i n F i g . 3.  The a l g e b r a i c values of the p r i n c i p a l s t r e s s e s i s assumed t o have the f o l l o w i n g r e l a t i o n , i n which the t e n s i o n i s taken p o s i t i v e and compression negative.  <^ > x  >  The maximum shear theory s t a t e s t h a t y i e l d i n g begins when the maximum shearing s t r e s s beoomes equal t o the maximum shearing s t r e s s at the y i e l d p o i n t i n simple t e n s i o n .  Sinoe  the maximum shearing s t r e s s i s equal t o h a l f the d i f f e r e n c e between the maximum and the minimum p r i n c i p a l s t r e s s , the  8  condition f o r y i e l d i n g i s  IT*  The shearing s t r e s s (6) to the ^  plane may  6*$} - I** ^ ' f on a c r o s s - s e c t i o n p a r a l l e l andT*^ ,  be d i v i d e d i n t o two components m  p a r a l l e l r e s p e c t i v e l y t o the ^ and £ axes.  In t h i s method of  designation of s t r e s s components, the s i n g l e s u b s c r i p t of a normal s t r e s s suoh as J*, and the f i r s t subscript of a s t r e s s suoh as  shearing  correspond w i t h the d i r e c t i o n of the normal  to the s e c t i o n , while the second s u b s c r i p t o f ' T ^ j the d i r e c t i o n i n which the oomponent i s to be  indicates  taken.  The values of the p r i n c i p a l shearing s t r e s s e s 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 . mum  For the general state of s t r e s s , the maxi-  shearing s t r e s s occurs a t sections making angles of 4 5  degrees w i t h the s e c t i o n s across which the p r i n o i p a l s t r e s s e s act. Timoshenko ( 5 ) desoribes a simple procedure to estimate the f a t i g u e strength f o r d i f f e r e n t combinations of v a r i a b l e and steady loads.  This method i s based on:  (a) the maximum shear theory f o r d u c t i l e m a t e r i a l s (b) the y i e l d s t r e s s f o r steady s t r e s s e s (c) the endurance l i m i t f o r v a r i a b l e s t r e s s e s (d) s t r e s s concentration i n d u e t i l e m a t e r i a l s i s important f o r v a r i a b l e loads only. (e) the assumed law of f a i l u r e f o r oombined steady and v a r i a b l e s t r e s s e s i s shown i n F i g . 6 and i s conservat i v e compared t o (tough's experimental r e s u l t s .  a  A S S U M E D " <5AFC U t M l f UlNE. ASSUMtO l_AW  OF  FMLUR.E  z  t  • A L A W . o r FAILURE  R<s. 7  COM&INED 5 T R £ S S PLANES  So  Se • endurance l i m i t f o r reversed  II  stress  ( f a i l u r e c o n d i t i o n f o r complete load r e v e r s a l ) Sy - y i e l d point s t r e s s i n tension n  • f a c t o r of s a f e t y  The equation of the f a i l u r e l i n e i n terms of the steady s t r e s s So and the c y c l i c s t r e s s Sv i s : So Sy  Sv Se"  _ J _  ^  s  . 1  I f we introduce n, the equation of the safe l i m i t l i n e i s : So Sy  Sv _ Se "  i  5;  Applying these assumptions t o s h a f t 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 s t r e s s  _ , ,„ _ (Ss)o = T o d . 1 6 T 0 '  due to To  2 J  shear s t r e s s due  to  (Ss)v =kTvjd  Tv  2J  rrd  s k i 6 Tv -77-</ 3  k • s t r e s s concentration f a c t o r due t o a f i l l e t , keyway, e t c .  Since  (Ss)o (Ss)y  . ^  I6T0  4.  77"J (Ss)y 3  VW  (Ss)v = 1 (Ss)e n K16Tv 3  (Ss)e  = ! n  (Ss)y * §Z 2 Henoe  Case 11  d = ~ * rr  (Ss)e = Se 2  I*  (To kTv)' (Sy Se) +  a steady moment Mo and a c y c l i c moment Mv loading. = Mo"t Mv s i n wt  normal s t r e s s So • Mod _ 52 Mo due  to  Mo  2  normal s t r e s s due t o Mv  TTo/"  3  Sv = KMv cl a k52Mv 2  Since  1  1  So . Sv - 1 Sy Se n 52 Mo 3 Sy  W Henoe  k32 Mv 3 Se  -  ^  1  n  d = /»§& (Mo^kM^J / 7T (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 w t ) + ( T o t T v s i n wt) In t h i s case we have combined normal and shear s t r e s s e s which are shown i n F i g . 7 f o r a given plane P. P.  normal s t r e s s due Sx -  . . , „, . & kMv d  M o  to Mo^ Mv  +  2 1  shear s t r e s s due  m  Sxy  a  to To - Tv  2 1  , „ To d ± kTv d m  2 J  2 J  Ss = combined shear s t r e s s dm plane P. P. = AC s i n 2(0+-A) » Sx s i n 2A •+• Sxy cos 2A 2 Ss = (Mo d ... kMv d) s i n 2A _L. (Tod kTvd) oos 2A ( 41 41 ) ( 2J 2J )  Hence  +  (Ss)o « Mo d s i n 2A 41  To d cos 2A ZZ  (Ss)v • kMv d s i n 2A_i_ k Tv d 41 2J  cos 2A  Substituting 1 = n 1 = (Mod s i n 2A n ' 41  Tod oos 2A) 2J  (Ss)y  (kMvd s i n 2A 41 -  kTvd cos 2A) 2J  (Ss)e  Hence 3  & s/52n(Mo s i n /  7T  (—  sy  To oos 2A  kMv s i n 2A-+-kTv oos 2A) Se )  where (Ss)y = §L , 2  (Ss)e = Se ; 2  t a n 2A = 1/2 (Mo+k Mv) (To+k Tv)  ANAYLSIS OF A KEYED PROPELLOR ASSEMBLY  The f a t i g u e strength of a keyed p r o p e l l o r 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 t h r u s t stresses f o r i d e a l c o n d i t i o n s of l o a d i n g , that i s , calm seas and a f u l l y submerged p r o p e l l o r . are a number of indeterminate  But there  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 p r o p e l l o r i s r a c i n g , the maximum bending s t r e s s e s when the ship i s i n b a l l a s t , and the clamping s t r e s s e s a t the top of the taper by the edge of the p r o p e l l o r 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 s o l u t i o n f o r the s t r e s s conc e n t r a t i o n a t a keyway does not e x i s t , but the "hydrodynamloal analogy" ( 5 ) i s u s e f u l f o r d i s o u s s i n g the t o r s i o n of s h a f t s . The t w i s t i n g of shafts of uniform c r o s s - s e c t i o n i s mathematic a l l y i d e n t i c a l t o the motion of a f r i c t i o n l e s s f l u i d moving w i t h uniform angular v e l o c i t y i n s i d e a s h e l l having the same s e c t i o n 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 a t  any p o i n t i s taken as representing the shearing s t r e s s a t that point of t h e cross s e c t i o n of the bar when t w i s t e d . For the case of a keyway w i t h sharp corners (See Fig.9) t h i s analogy i n d i o a t e s 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 a t the outside corners  (points M-M); hence the  shearing s t r e s s i n the corresponding t o r s i o n problem i s zero, at such p o i n t s .  S i m i l a r i l y , t h i s analogy i n d i c a t e s an i n f i n i t e  v e l o c i t y a t the i n s i d e corners (points N-N), the v e r t i o e s of  I  I i  I I  "UbERTY*  SHIP  the reentrant corners; henoe the shearing s t r e s s i s a l s o i n f i n i t e at suoh points.  Sinoe a keyway w i t h  absolutely  sharp corners cannot be cut, reentrant F i g . 9 Hydrodynamioal Analogy f a c t o r K" W  the  corners w i l l have a small  r a d i u s of f i l l e t and a f i n i t e s t r e s s distribution.  A stress  concentration  i s used t o denote the r a t i o of the maximum s t r e s s to  the nominal s t r e s s ; that i s ,  K =  ' nom.  m a x  o-*  F i g s . 10 and 11 show published values of the s t r e s s concentrat i o n f a c t o r "K" as have been obtained by the soap f i l m method and the p h o t o e l a s t i c method of measuring s t r e s s . Several experimenters have c a r r i e d out endurance t e s t s to determine the nominal s t r e s s concentration  factors  f o r sled-runner and round-ended keyways, but because of s i z e e f f e c t and clamping s t r e s s S. Archer (1) maintains that u n t i l l a r g e - s c a l e t e s t s of f u l l - s i z e p r o p e l l o r s are made, there seems to be l i t t l e t o choose between the two types of keyways. Under steady conditions i n smooth water, w i t h some bending f a t i g u e present, such as when the p r o p e l l o r i s p a r t l y submerged i n the b a l l a s t e d conditions, i t i s estimated from the evidence ( l ) so f a r a v a i l a b l e that the f a t i g u e strength of a w e l l f i t t e d p r o p e l l o r assembly, expressed i n terms of nominal reversed t o r s i o n a l s t r e s s , 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 ed value may  estimat-  be s e r i o u s l y reduced i n s e r v i c e on aooount of a  v a r i e t y o f oauses, such as extremes of s t r e s s  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= T  N Q M  ,WHERE ^NOM. =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  a r i s i n g from machining erros, badly f i t t e d parts, corrosive envirnment, and previous overstress. Stress concentration at a keyway (7) i s best r e a l i z ed with the a i d of photoelasticity (8) which u t i l i z e s the o p t i c a l effeot of stresses on a transparent model.  Certain  transparent materials (9) when stressed become birefringent, that i s i f a beam of l i g h t be passed through the stressed material i t i s s p l i t up into two plane polarized waves of l i g h t which vibrate i n planes at r i g h t angles to each other.  At each  point i n the material the two planes coincide with the d i r e c tions of the p r i n c i p a l 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 p r i n c i p a l stresses and on the thickness of the material.  Thus, on emergence, the two waves  are retarded r e l a t i v e to one another, or out of phase, by an amount proportional to the difference of the p r i n c i p a l stresses. To observe the r e l a t i v e retardation i t i s necessary to resolve the two component waves i n one d i r e c t i o n .  This may convenient-  l y be done by using an analysing unit i n the emergent beam of light.  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 i b l e i n the form of fringes on a suitable screen.  Therefore by using a two dimensional model, the r e s u l t -  ing fringe pattern i s an indication of the p r i n c i p a l stress  No Load  Load - 2 l b s .  Load - 4 l b s .  Load - 5 lbs.  F i g . 14 Stress Patterns i n a Keyed Shaft assembly  d i f f e r e n c e a t a l l p o i n t s i n the model.  By s u i t a b l e c a l i b r a t i o n  the s t r e s s d i f f e r e n c e per f r i n g e oan be determined. The w r i t e r made the keyed shaft assembly shown i n Fig.  12 and s t u d i e d i t i n the p h o t o e l a s t i c apparatus of the  Mechanical Department shown i n F i g . 15. The s t r e s s concentrat i o n s a t the reentrant corners are e s p e c i a l l y obvious i n Fig.14 and demonstrate the d e t r i m e n t a l e f f e c t of keyways.  Some r e v i s i o n s i n the keyway design of " L i b e r t y * 1  t a i l s h a f t s have been made t o increase t h e i r f a t i g u e s t r e n g t h . The o r i g i n a l keyway as shown i n F i g .  15, a t r a c i n g of the  Burrard Dry Dook s p e c i f i c a t i o n s of 1943, was of the roundended type w i t h no r e g u l a t i o n s governing the f i l l e t .  I n 1947,  Mr. W. 0. Richmond (15) recommended a sled-runner keyway w i t h 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 r e v i s e d keyway as shown i n F i g . 16, a t r a c i n g of the Baldwin Looomotive Works s p 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 w i t h a 5/16 i n . f i l l e t and has i t s edge broken w i t h a f i l e . The key has a 1/4 i n . chamfer on a l l s i d e s and i s t o be held i n p l a c e by two jack-screws.  I n a d d i t i o n , the key has 1/8 i n .  saw cuts a t the forward corners t o t r a n s f e r the load more equally throughout i t s l e n g t h . Furthermore, t a i l s h a f t s are t o be drawn and inspected every two years r a t h e r than t h r e e .  24  FATIGUE TESTING MACHINES  Economy and machining d i f f i c u l t i e s l i m i t e d the t e s t bar to a 3/4  i n . diameter shaft w i t h a 3/16  i n . keyway.  Using  t h i s shaft s i z e as a b a s i s 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 reversed s t r e s s e s .  t o repeated  In a l l t e s t s , a known load was a p p l i e d and  the number of s t r e s s c y c l e s was measured w i t h a counter.  Since  these machines were constructed as p r e l i m i n a r y models, a l i s t of suggested improvements i s given i n the Recommendations. The w r i t e r wishes t o acknowledge t h a t 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 U n i v e r s i t y of I l l i n o i s  (10),  and the e c c e n t r i c of the t o r s i o n machine resembles one designed by H o l l e y (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  d r i v e n by a motor and the other end contains a tough s t e e l bushing which supports the t e s t s h a f t .  The r o t a t i n g t e s t bar  mounted as a c a n t i l e v e r beam, i s loaded t r a n s v e r s e l y w i t h 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 l a r g e end of the tapered cone, which i s the o r i t i c a l s e c t i o n i n l o a d i n g , has shearing s t r e s s e s as w e l l as  F i g . 18 Reversed Bending Machine - General Arrangement  6>!LL  O F MATE1PJ/M_  T  No. I  R r F E . R H . U C E L .  TU&E  B Cb.uPu M<£. !  4 " £? BRACKET  7  BRACKET"  8  ; *3i f  & O L "  'H  W.ASH.E.PL  "I  15  .i  R E A M E D H O L E S F O R . ITEM "3 3/ " "4 O ' D - ' S T ^ - ^ R*&»**8  T I 8  ii 1 —  3 artta  ift^r i.  if  11  en. -t;  I  5%*  ^  1;  i " M:S-. T U B E .  M A C H I N E To  -1  i.2: 2 - M- 5-  BcAfciMq  L  33. 9  (5  HP^ j (' _i  —i—  id-  2'  i - M - S - WASHER. M K I  i •e-  H O L E IN M K . F ~  1- T t  ST  O A R  3LOT IN  MicGj  IN  . \  ''4-' 5 E T bCRJflVN'S  /4  3  ^ " K . D  "2_  CASES. ,MK. .ft  (  Frr  HK.. A  \  \  A.  .8  1" M ' 5 - COUPLING, H i C  ; - M - S - BOLT M K . H '  HI P p ^ S S F\l  §1  / ... •• t  13  te  -  SMTO M K - A  i  LJ_j.  iff ±  i  i i  i  | 4.1-5T-S'  r  t  i—•r  >  f  COLLAR. M O  '6 j 1 - M - S -  BRACKET-  MtcF  (7  BRACKET  MKJ'G  1-M-S-  MECHANICAL ENQINEE.R.1.NG UNIVERSITY OF &R.ITISH COLUMBIA  REVER.SED.BENDING MACHINE MAY  SCALE.. D R A W H 6X  UT"C  1^4^  DRAWING  NO.  7-1  the maximum bending stresses a c t i n g on i t . concentration f a i l u r e there.  However, the s t r e s s  a t the forward end of the keyway u s u a l l y causes The nominal stresses were c a l c u l a t e d f o r each  specimen and are l i s t e d i n Table I . As the maohine r a n smoothly f o r a l l t e s t s , no m o d i f i cations were necessary.  REVERSED TORSION MACHINE The reversed t o r s i o n maohine c o n s i s t s mainly of the t o r s i o n arm which t w i s t s one end of the t e s t bar, while the other end i s held f i r m l y i n the t o r s i o n block. the arm t o the e c c e n t r i c p l a t e .  A l i n k connects  By a d j u s t i n g the s l o t t e d  e c c e n t r i c p l a t e , the angular displacement of the arm and hence the applied torque, oan be v a r i e d .  The e c c e n t r i c i s mounted on  a shaft d i r e c t l y connected t o a 3/4 hp Baldor motor; f o r d e t a i l s of t h i s motor r e f e r t o (12).  variable-speed F i g . 20 shows  the general arrangement of the maohine and F i g . 21 shows a view of the e c c e n t r i c .  The d e t a i l s of the maohine are g i v e n i n  F i g . 22. The maohine was fastened t o the concrete f l o o r w i t h anohor-bolts t o 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  A l s o , the running speed was reduced from  1800 t o 1000 rpm t o l e s s e n wear a t the l i n k - e c c e n t r i c connection.  However, a l i t t l e movement which was noticed i n the  t o r s i o n block was assumed due t o f l e x i b i l i t y i n the maohine itself.  28 TABLE 1  Number 1 '  REVERSED BENDING DATA  Keyway  Bending Stress  SR  «  Shear Stress  Number o f Cycles  Remarks  13600  113  22624000  UNBR  21700  171  507000  BR BR  2 •  SR  16300  128  5012000  ... .3 '  SR  14000  131  iq0417000_  16800  156  2204000  BR  .  UNBR  4  RE  16200  128  1044000  BR  5  RE  13600  128  3430000  BR  ;  6  RE  10800 _ 13600  85 106  UNBR BR  , |  30577000 j 38180~00 T  SR - sled-runner  keyway  RE m round-ended  keyway  BR  broken  s  *  UNBR B unbroken  Bending S t r e s s r MP. Shear S t r e s s  A  blLL  ECC&NTWC PLATE  3  -  I 9  -  1  far&qmt hue  ; Z . AB  U  REFE.^£:NCE.  QUAKL  PART  i  1  MATE1WAL.  n  No.  a  OF  ,  MK.  1  "TOBSXOTH B L O C K . ;  1  i&  •£>E1AR-\M^.  2.  "0" "E.  BEARJNG,  s  10  "A°  ."foPSvON A R M  i  / 4 - ff  I 1  , L  "p  u  10 8 9  3  3  xi8" S H A F T  IO  PLAN  MOTO^  J J _[ '/a>  1 f*•  bouT  12 •  =3  DETAIL O F  ft  CCCE.NTR4C  8  3  4  -  .'7  m  Lt  E L ELVATION  END  EJ  74-  View  V% + REAMED  ®5  i IS' P^.  ^4-'  REAM  to  S L O T  4- K f l Y W A f  ®  2£  l - M ' 5 - TORS\OM Pm^  H^F  S^URSCJE:  i  «  L  KG  12'  SHOULDER.  iff!  •7% II 7  1  1 - M - 5 • E O E K T W C PLATE:  (% 1.-M-5-  A  ECCENTWC  _1L±15- TEST  &A*.  \\  MK.Q.  Hub M*Jd ^ ' REAMED  4"  u §T  -»  6'  9  §s—; o  8  l - H - 5 - LINK.  MK.H  V  % DP-ILL^  3  I'M'5'  JORS\ON  ftLQOc  MKJ'C  1  4  1-H-S- 5 E A R t N ^ - ' H l J D '  i  1-H-S- B & A R J N Q MK."E"  MECHANICAL E N G I N E E R I N G U N I V E R S I T Y O F ftHiTiSH C O L U M B I A  D R A W N S>Y~~ O T " C. aSxtDJ  DRAWING  FlQ.22  ^40.  HlllHimi^-^H  F i g . 23  SR-4 Strain Indicator  34  F i g . 24  Brush Oscillograph  —  3Z  The s t r a i n and hence the stress i n 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 t h i n paper c a r r i e r they were cemented on the edges of the torsion arm and covered with wax as shown i n F i g . 21.  When the arm was displaced, one  of the Gages was strained i n tension and the other i n compression.  The change i n resistance which i s proportional to the  change i n s t r a i n , can be measured and converted to stress. Therefore, knowing the s t r a i n i n the torsion arm, the stress i n a specimen could be calculated from the c a l i b r a t i o n curve shown i n Appendix A. The s t a t i c strains (when the machine was stopped) were measured with the SR-4 Strain Indicator shown i n F i g . 23.  The dynamic strains (when the machine was running)  were measured with the Brush Oscillograph as shown i n 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 experimenta l e r r o r , the o s c i l l o g r a p h curves were used to measure the applied torque. A l s o , these nominal s t r e s s e s agreed w i t h the a c t u a l s t r e s s e s i n the t e s t shaft measured w i t h a separate s t r a i n gage shown i n F i g . 25.  A complete d e s c r i p t i o n of the  measuring instruments i s given i n a paper by Johnson and Bruce (12).  The nominal stresses were c a l c u l a t e d f o r 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 t e s t s h a f t when the two bearings supporting the t o r s i o n s h a f t were removed.  The t o r s i o n a l  stresses were c a l c u l a t e d from the a p p l i e d torque and the bending stresses were measured by s t r a i n gages on the t e s t s h a f t . The nominal stresses were c a l c u l a t e d f o r each specimen and are shown i n Table I I I . Graphs of the o y o l i c s t r e s s e s as measured by the Brush O s c i l l o g r a p h are shown i n F i g . 26.  REVERSED TORSION DATA  TABLE 11  Number  Keyway  1  RE  2 4 5 6  SR SR RE SR  Torsion Stress 8500 9400 9400 8400 8000 8000  Number of Cycles 1022E00 261700 230000 275000 312000 1163500  SR s sled-runner keyway RE s round-ended keyway BR r broken UNBR = unbroken  Torsion S t r e s s =  ~  Remarks UMBR BR BR at f l a t s BR at f i l l e t BR BR  j »  3S  TABLE 111  Number  -  COMBINED REVERSED BENDING AMD TORSION DATA  Keyway  3 7  SR RE  8  SR  Bending Stress 60000 3300  Torsion Stress 4000 9400  3300  Number of Cycles 8700 . 14100 41000  9400  SR a sled-runner keyway RE - round-ended keyway BR m broken UNBR = unbroken  Bending S t r e s s = Torsion S t r e s s =  ~  Remarks BR BR BR  36  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  OBSERVATIONS  1.  37  A t f i r s t the bearings supporting the e c c e n t r i c s h a f t heated  up and threatened applied.  t o s e i z e when the maximum e c c e n t r i c i t y was  On the advice of Professor Wm. Wolfe, S u l f u r Flowers  was added to the o i l .  Immediately the bearings cooled t o a  moderate temperature and the wear from then on seemed negligible.  Therefore, S u l f u r a c t s very w e l l as an extreme  pressure l u b r i c a n t , f o r s t e e l s h a f t s i n bronze bushings.  2.  A l l of the s h a f t s tested i n the t o r s i o n machine, e s p e c i a l l y  ones w i t h sled-runner keyways heated up t o a temperature t h a t could hardly be touched.  Since t h i s i s due t o the work done,  i t may have some e f f e o t on t h e f a t i g u e strength as the s h a f t s were h o t t e r f o r higher a p p l i e d s t r e s s e s .  RESULTS  1.  I n a l l oases the sled-runner keyway withstood more c y c l e s  at a given s t r e s s than the round-ended keyway.  2.  Reversed Bending F r a c t u r e s are shown i n F i g . 27. (a) the sled-runner keyway showed a c i r c u m f e r e n t i a l orack on the d r i v i n g s i d e 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 s e r i e s 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 a t the root of the forward corners and progressed a t 45 degrees to the s h a f t a x i s .  These cracks became v i s i b l e a t  about 500,000 c y c l e s and the w r i t e r watched them progress u n t i l f a i l u r e a t 1,163,500 oyoles.  This  specimen f a i l e d e x a c t l y as p r e d i c t e d by the maximum shear theory. (b) the round-ended keyway had a r a d i a l crack s t a r t a t each s i d e of the forward end a t approximately 45 degrees t o the s h a f t a x i s .  4.  Combined Bending and Torsion f r a c t u r e s are shown i n F i g . 29.  Both types of keyways had cracks a t the forward end and showed some evidence t h a t the angle of f r a c t u r e 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  5.  The  30 had  stresses.  reversed  t o r s i o n f r a c t u r e a t a f i l l e t as shown i n F i g .  a c i r c u m f e r e n t i a l crack at the  cracks p o i n t i n g towards the  stress r a i s e r with r a d i a l  centre.  •Fig. 30 Reversed T o r s i o n F r a c t u r e  6.  S u l f u r was  e f f e c t i v e l y used as an  cant f o r s t e e l on bronze.  at a  Fillet  extreme p r e s s u r e  lubri-  43 CONCHTSIOHS  1.  A sled-runner keyway has a higher f a t i g u e strength than a  round-ended keyway i n a keyed tapered-shaft assembly stressed by reversed bending, reversed t o r s i o n , or combined reversed bending and reversed t o r s i o n .  2.  The maximum shear theory of strength p r e d i c t s the plane of  fatigue f a i l u r e . (a) bending stresses produce a c i r c u m f e r e n t i a l f r a c t u r e p e r p i n d i c u l a r 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 f r a o t u r e 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 f r a o t u r e whose angle of i n c l i n a t i o n to the shaft a x i s r  i s governed by the combined s t r e s s .  3.  Owing to s i z e e f f e o t , the d i f f e r e n c e i n f a t i g u e  between the two types of keyways may  strength  be l e s s 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 t o r s i o n machine. 5.  Since l i t t l e research has been done on the s t r e n g t h  of  keyed connections, these machines could be u s e f u l f o r f u r t h e r i n v e s t i g a t i o n s on t h i s  subject.  RECOMMENDATIONS  1.  R o l l e r bearings would be s u i t a b l e f o r t h e two b e a r i n g s  s u p p o r t i n g t h e e c c e n t r i c s h a f t r a t h e r than t h e bronze  2.  I f the t o r s i o n arm was reduced  sleeve.  t o 10 i n . of e f f e c t i v e  l e n g t h , h i g h e r speeds and h i g h e r s t r e s s e s c o u l d be used; o r f o r the same c o n d i t i o n s , smoother o p e r a t i o n would r e s u l t .  3.  F o r an exact d e t e r m i n a t i o n of the f a t i g u e s t r e n g t h of  keyed connections, the s h a f t and the bushing should be ground to s i z e .  Vancouver E n g i n e e r i n g Works has t h e equipment t o do  t h i s job.  4.  By t h e use of s u i t a b l e bushings, b o t h machines oould be  m o d i f i e d t o t e s t s t r a i g h t keyed connections o r p r e s s  fits.  45  APPENDIX A  The SR-4 S t r a i n Gages which measured the a p p l i e d torque on the t o r s i o n arm were c a l i b r a t e d by s t a t i c loads using the Baldwin SR-4 S t r a i n I n d i c a t o r . s i s t s of a four arm Wheatstone bridge.  This i n d i c a t o r con-, 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 s t r e s s changes the gage r e s i s t a n c e s . When a gage i s s t r a i n e d i n tension i t s length increases and i t s diameter decreases w i t h a r e s u l t a n t 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" t o 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 f a c t o r F.  The u n i t s t r a i n i s equal t o  the u n i t r e s i s t a n c e change d i v i d e d by the gage f a c t o r .  This  i n d i c a t o r 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 t o r s i o n arm to be l i n e a r as expected.  46  CALIBRAT ION OF TOR3I0N ARM ST RAIN GAGES F i g . 31 CQ <D  . C C C 1  c >> c)  / -  -  c>  <s  >  •sion Arm ( Indioa- ior Differ* >nce On Toi r licroinche!  w>  <3UU  u l rt  100 -  0  3000  12 D00 60 00 90 DO Shear Stress i n rest Shaft psi.  15 300  47  APPENDIX B  To show that the s t a t i c and dynamic s t r e s s e s i n a t o r s i o n t e s t shaft agreed w i t h i n experimental e r r o r .  The  s t a t i c s t r a i n s were measured w i t h the SR-4 S t r a i n I n d i c a t o r . The dynamic s t r a i n s were measured w i t h the Brush O s c i l l o g r a p h .  Nominal Stresses The nominal s t r e s s e s were c a l c u l a t e d f o r the a p p l i e d 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  e  B  strain  r  500  " H  micro-inches  From the c a l i b r a t i o n curve i n Appendix A SI  - ± 13600 p s i .  (b) Dynamic Load  E-t- AE  V - 6 volts  48  From i n i t i a l s e t t i n g 15 mm d e f l e c t i o n = 5 m i l l i - v o l t s 7 mm  AR R  -  2AE V  a  1 F  e  "  2.33  8(.00235) 6  10078  ohms per ohm  1_ (.0078) _ .000380 inches 2.06  AR R F  "  s  gage f a c t o r  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 s t r e s s e s were c a l c u l a t e d from the s t r a i n observed i n a s t r a i n gage a t 45 degrees t o shaft a x i s as shown i n F i g . 25. (a) S t a t i c Load Maximum Reading  -  Minimum Reading  s  815  "  =  1170  »  Difference  e  1985 micro-inches  585 micro-inches  e i13500 p s i  (b) Dynamic Load v  E  1  AR  V = 2i -  small  V = E+A E  AE  V 2  V AR 4 R  AE = From i n i t i a l  (R+«R)  =  V IR-HAR) 2R+AR  =  6 volts  setting 15 mm deflection = 5 m i l l i - v o l t s 5.5 mm  AR  "  S, '  H  = 4 AE = 4 (.00183) = . —y— g— .00122 ohms per ohm - 1 AR 1 (.00122) = F ~R 2706 " • inches J  e  • 1.83  O Q  u  =  = I_JL_ = ~ 1+ m  0 0 0 5 9 0  (30 x 10 ) (590 x 10" ) 1.3 6  6  -13600 p s i  s +  The wire from the s t r a i n 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.  SO  BIBLIOGRAPHY  1. Archer, S., Screwshaft C a s u a l t i e s - The Influence of T o r s i o n a l V i b r a t i o n and P r o p e l l o r Tmmersion, London, The I n s t i t u t e of Naval A r c h i t e c t s and Marine Engineers, A p r i l 8, 1949.  2. S e i t z , F r e d r i c k , 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, P a r t 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 M a t e r i a l s . New York, D. Van Nostrand Co., 2 v o l s , 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. S o l a k i n ,  and K a r e l i t z ,  " P h o t o e l a s t i o Study of Shearing  Stresses i n Keys and Keyways". ASME, 1931. 8. Frocht, M. M., P h o t o e l a s t i o i t y , New York, John Wiley & Sons, 2 v o l s , 1941.  5/  9. Shaw, F. S., "Determination of S t r e s s Concentration F a c t o r s , " A Symposium of the Fatigue of Metals. Melbourne, U n i v e r s i t y of Melbourne,  1947.  10. Moore, H. F., and Krouse, G. N., Repeated-Stress  (Fatigue)  Testing Machines Used i n the M a t e r i a l s Testing Laboratory of the U n i v e r s i t y of I l l i n o i s . Urbana, 111. Exp. S t a . , C i r c u l a r 23,  1934.  11. H o l l e y , E. G., The S t a t i c and Fatigue T o r s i o n a l Strengths of Various S t e e l s 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  P r o p e l l o r S h a f t i n g . Vancouver, B. C ,  1949.  13. Richmond, W. 0., F a i l u r e of T a i l Shafts of V i c t o r y 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. S t a . , Urbana, B u l l e t i n 42. 16. Dorey, S. F.. L i m i t s of T o r s i o n a l S t r e s s i n Marine O i l Engine S h a f t i n g . London, The Engineer, A p r i l 11,1947.  17. Gough, H. J . , The Fatigue of Metals. New York, D.  Van  MoBtrand Oo. L t d . , 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 P r o j e c t C y c l i c Stresses i n Ship P r o p e l l o r S h a f t i n g , Vancouver, B. C ,  February 12,  1949  5Z  

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