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The effect of the mean stress on the endurance limit Ukrainetz, Paul Ruvim 1960

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THE EFFECT OF THE MEAN STRESS ON THE ENDURANCE LIMIT by PAUL RUVIM UKRAINETZ B.E., University of Saskatchewan, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, I960 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. ABSTRACT The effect of the mean stress on the endurance l i m i t i s a matter of considerable academic and practical importance* So many variable and un-certain factors are involved that i t i s not surprising that many different formulas and theories have been proposed since the start of investigations into this subject i n about 1858. In this thesis, factual data obtained i n the course of tests planned primarily for the purpose of determining the effect of the mean stress are presented and discussed* A standard fatigue testing machine vas used for a l l the tests* Various mean stresses* both for axial-load and torsional tests, were employed* A c r i t i c a l examination of the proposed theories of fatigue failure has been made* The theory which considered that the inception of fatigue resulted from alternating shearing stress and that the resistance to fatigue was influenced by the magnitude and sign of the steady normal stress was found to explain best the damage done to the material structure* I t has been the opinion of a number of investigators that the fatigue strength under axial-load i s decreased by an increasing tensile mean stress and i n torsion the fatigue strength i s unaffected by the mean stress* The experiments done here clearly indicate that this conclusion i s true* - i i -TABLE OF CONTENTS Title Page No. Abstract i i Acknowledgment •<>••• v Introduction • . . . . . . . . . . . . . . . 1 Brief History of Earlier Work • • • • • 3 Apparatus • • • • • • • • • • • 9 Material and Specimens . . . . . . . . . . . . . . . . . . . . . . 15 Test Program 20 Test Results 22 Static 22 Fatigue 23 Correlation and Discussion of Test Results • • • • • • • 29 Conclusions and Recommendations • • • • • • • • • • • • • • • • • • 34 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 34 Recommendations • • 34 Appendix A. Calculation of Torsional Specimen Free Length • • • • 35 Appendix B. Verification of Machine Loads • • • • • • 36 Appendix C. Static Tensile Tests • • 38 Appendix D. Fatigue Test Data • • • • • »• 39 Bibliography • • • • • • 41 - i i i -LIST OF ILLUSTRATIONS Figure Page Ho. 1. Internal Mechanism of Fatigue Testing Machine . . . . . . . . . 12 2* Fixture For Axial Load Fatigue Tests . . . . . . . . . . . . . 13 3. Fixture For Torsional Fatigue Tests! . . . . . . . . . . . . . . H 4* Photomicrographs of Longitudinal Section of a Specimen . . . . . 17 5. Axial Tensile Fatigue Specimen • • • . . . . . . . . . . . . . 18 6* Torsional Fatigue Specimen . . . 19 7. Axial Fatigue Tests With Various Mean Stresses . . . 25 8* Effect of Tensile Mean Stress on the Fatigue Strength Under Axial Load 26 9* Torsional Fatigue Tests With Various Mean Stresses 27 10. Effect of Mean Stress on the Fatigue Strength in Shear Due to Torsion 28 - iv -ACKNOWLEDGMENT The author i s indebted to Professor W« 0« Richmond* who suggested this investigation and contributed time* effort, and ideas to i t * Appreciation i s expressed for the generous co-operation of Br* J* A* H* Lund i n making available the equipment of the Physical Metallurgy Laboratory used l n this investigation* The author gratefully acknowledges receipt of a Bursary granted by the National Research Council* Funds for the purchase of equipment and supplies were available from a National Research Council Grant-in-Aid of research awarded to Professor W* 0* Richmond* - 1 -THE EFFECT OF THE MEAN STRESS ON THE ENDURANCE LIMIT Introduction The endurance l i m i t of a material Is defined as the maximum alternating (completely reversed) stress that can be superimposed on the mean or steady stress and be repeated an indefinitely large number of times without causing progressive fracture (fatigue f a i l u r e ) . This value i s also frequently referred to as the fatigue strength for the given mean stress* Many Investigators have concerned themselves with the study of the effect of the mean stress on the fatigue strength and even a cursory survey of literature shows that this s t i l l remains an open question* Experimental evidence* however* appears to show that* i n the case of a tensile mean stress, the fatigue strength, defined i n terms of the alternating stress amplitude, decreases as the mean stress of the cycle increases to higher tensions* The rate of decrease i n fatigue strength with tensile mean stress depends apparently on the material and varies from slight to large with different materials* For a compressive mean stress, fatigue tests indicate a slight to large increase i n fatigue strength with increasing values of compressive mean stress* I f , on the other hand, a material i s subjected to torsional fatigue, i t has been shown i n a number of investigations that the fatigue strength i s independent of the mean stress, provided that the maximum shearing stress does not reach a value that causes the material to f a i l by general yielding* In other words, at the highest maximum shearing stress there i s no permanent plastic deformation of a rel a t i v e l y large portion of the specimen on the f i r s t application of load* I t may appear evident from the above statements that the matter of the effect of the mean stress on the endurance l i m i t i s well established* But as was mentioned earlier, this i s not so, for extraordinary anomalies persist and the proposed theories of failure show considerable variation* Therefore* the purpose of this investigation i s to establish some of the facts i n relation to one particular ferrous material* An axial-load type of machine was used for the tensile mean stress tests* This machine w i l l be described later* The same machine* by employing a special fixture* was used to accommodate the torsion tests* The data include results of tests at different values of mean stress i n tension and i n torsion* A comprehensive study of the test data i s made to determine the manner i n which the mean stress affects the fatigue strength* The machine had limitations as to the application of a tr u l y axial compressive load to the specimen and hence fatigue tests to determine the effect of a compressive mean stress on the fatigue strength were omitted* - 3 -B r i e f H i s t o r y o f E a r l i e r Work Fatigue tests on the effect of the mean stress were f i r s t started by Wohler 1 (1858-70) who concerned himself with mean tension only. The results of these tests were used by Gerber (1874) to formulate a general-ization on the effect of mean stress, this being the Gerber parabolic 2 relationship R/Ro s 1 - (H/U)2 (where R i s the semi-range of alternating stress at mean stress M, RQ the semi-range of alternating stress at zero mean stress, and U the ultimate tensile stress)* Although no tests with mean compression were carried out. Gerber concluded that since i n bending failure always occurred i n the tensile fibers, the endurance l i m i t would be higher i n compression than i n tension and that to use tensile results as valid for compression would give answers on the safe side* This was the basic reason for his suggesting a symmetrical form such as a parabola to re-late the mean stress with the fatigue strength* A few years later Bauschinger (1886), having accepted the Gerber relationship dogmatically, used i t with the aid of a few endurance tests to obtain curves for the behavior of several materials* His results, however, have since been disproved and his contribution to this f i e l d can thus be n u l l i f i e d * In 1914, the modified Goodman law R/RO = 1 - M/U was devised by Goodman3 based chiefly on test results by Wohler* In the diagram representing this law. the mean stress i s expressed as a function of the ultimate tensile stress* Soderberg^ (1930) proposed a s l i g h t l y different relationship called the Soderberg line 4 -R/Ro S l - M/Y where the ultimate tensile stress i s replaced by X the yield stress* It is desirable at this point to indicate that none of the empirical relations that have been proposed to describe the observed effect of mean stresses i n fatigue can be used to relate the behavior of a l l materials* The Soderberg line has been found safe for nearly a l l materials and hence i s used most widely; but in many instances this line seriously over-estimates the effect of mean stress, and in a few i t underestimates i t * A summary of the work in this area up to 1942 i s contained in a paper by Smith''. The results of the various investigators were analyzed both for ferrous and non-ferrous materials* It was concluded in this paper that the endurance limit decreased i n general with increase i n steady tensile stress in accordance approximately with the modified Goodman law* When the mean stress was compression, i t was found that the endurance limit was approximately constant for a l l ranges of stress. The analysis here included results from axial tests only* The author is i n doubt as to the validity of the mean compression results, for i t seems that insufficient care had been taken to ensure a truly axial compressive load. Findley, Mergen, and Rosenberg^ (1953) ran a number of fatigue tests i n bending and their observations were that an increase in the mean stress caused only a small change i n the fatigue strength until yielding occurred, at which stress the strength generally decreased* The redistribution of stress, which inevitably results from even the most insignificant plastic flow* was corrected for. but these corrections are of extreme difficulty, and none of the methods employed give exact values of the stresses. This is quite understandable since a correction does not allow for yielding that might not be homogeneous or continuous, for the fact that the specimen shape confines yielding to a small region which may further complicate the geometry of plastio straining, and for the difference in the rate of straining during fatigue and static tests. Thus i t follows that bending tests do not give a true picture of the fatigue properties of a material; rather, the only type of test that wi l l give rise to no uncertain redistribution of stress is an axial one (push-pull test)* Having recognized this, Pindley'' (1954) performed axial-load fatigue tests using the same material as in the bending tests* It was observed this time that the fatigue strength decreased more rapidly with tensile mean stress. At compressive mean stress the fatigue strength increased substantially. Morrison and 01Connor1 in 1956 investigated the mean stress problem using a specially designed push-pull machine which was conclusively shown to deliver a truly axial load to the specimen both in tension and com-pression* At moderate values of mean stress they found that the fatigue strength decreased linearly as the mean stress was changed from compression to tensions Recently Sines 8 (1959) critically selected the data of a number of investigations and these data also show a linear relation between the mean stress and fatigue strength in the region where maximum stress does not exceed the yield strength* The tests as presented were chosen by Sines because the testing methods used ensured true axial loading* The e f f e c t o f mean s t r e s s on the t o r s i o n a l f a t i g u e properties o f a ma t e r i a l has received l i m i t e d a t t e n t i o n i n comparison with the e f f e c t of mean s t r e s s on the a x i a l - l o a d f a t i g u e p r o p e r t i e s * Perhaps the f i r s t concentrated attack on t h i s problem was made by Smith? i n 1939* Then i n 1942* Smith'' made a rather thorough study o f a l l a v a i l a b l e t e s t data and the r e s u l t s i n d i c a t e that the mean str e s s has no e f f e c t on the fatigue strength as long as the maximum str e s s o f the cyc l e does not exceed the t o r s i o n a l s t a t i c shearing y i e l d strength of the material* Even when the maximum s t r e s s does exceed the s t a t i c y i e l d the f a t i g u e l i m i t does not decrease r a p i d l y with increase i n mean s t r e s s * However* a l l the t e s t s t h a t contributed to t h i s conclusion were done on s o l i d specimens and no attempt has been made to minimize the possi b l e e f f e c t o f s t r e s s gradient* The r e s u l t s quoted* e s p e c i a l l y those near the s t a t i c e l a s t i c range, are thus open to severe c r i t i c i s m * The problem o f s t r e s s r e d i s t r i b u t i o n has already been discussed f o r bending and* needless to say* the same s t o r y holds i n the case o f tor s i o n * To reduce the e r r o r which r e s u l t s from t e s t s on s o l i d s p e c i -mens when the y i e l d i s exceeded a t some point* the author f e e l s that only t h i n tubular specimens should be j u s t i f i e d f o r use* and that s o l i d specimens are unsuitable f o r i n v e s t i g a t i o n i n t o the e f f e c t o f mean st r e s s on the t o r s i o n a l fatigue strength* I n 1953* Findley* Msrgen, and Rosenberg^ c a r r i e d out t e s t s a t mean stresses which produced y i e l d i n g i n s o l i d specimens* and they allowed f o r the r e d i s t r i b u t i o n o f s t r e s s due to y i e l d i n g * The r e s u l t s obtained at two mean stresses other than zero i n d i c a t e , i n one case* a small increase i n the fatigue strength and* i n the other, a decrease of the - 7 -sane amount* This rather erratic variation suggests that the correct-ions were inadequate and so are generally unreliable* More recently Ghodorowski 1 Q (1956) has published the results of torsional fatigue tests on hollow specimens* As f a r as i s known this i s the only experimental work on hollow specimens that has been done up u n t i l now* The results show that the fatigue strength i n shear i s affected by even a small mean stress* A tentative conclusion was drawn that i n the region investigated the fatigue l i m i t decreased linearly with increase i n mean stress* This i s certainly contrary to the findings of ear l i e r investigators* In an a r t i c l e by F i n d l e y 1 1 i n 1959, a relationship i s derived which shows the fatigue strength i n torsion as being nearly independent of the mean stress* However, after a reappraisal of the theory, he suggests that i n general the fatigue strength tends to decrease s l i g h t l y with increasing mean stress. I t i s convincingly evident that there i s a confusion of ideas on torsional fatigue and i t w i l l be interesting to note where the results of the present investigation w i l l f i t i n * I t should be noted at this point that only the work concerned with unnotched ductile metal specimens has been reviewed here* The mechanism of fatigue failure has received much discussion and various theories have been proposed. A. theory of fatigue failure proposed 12 by Orowan i n 1939 was based on the concept that fracture starts when a c r i t i c a l stress i s reached i n an inhomogeneity or when the numerical sum of both positive and negative pla s t i c strains reaches a c r i t i c a l value* This theory predicted that the fatigue strength was independent of the mean stress as long as the elastic l i m i t was not reached* The same - 8 -conclusion was reached by Freudenthal 1 3 (1946) i n a theory of fatigue failure based on the s t a t i s t i c a l aspect of fatigue* The theory which seems to agree closest with experimental evidence 6 was proposed i n 1953 by Findley, Mergen, and Rosenberg . I t was postu-lated that the alternating shearing stress was the prime factor causing the phenomenon of fatigue fracture and that the resistance to this action was influenced by the normal stress acting on the plane of the maximum shear stress* That i s , the normal stress on the plane of the maximum shear stress may tend to Inhibit s l i p , i f compression, or make s l i p easier, i f tension* Since the normal stress i s dir e c t l y affected by the mean stress, i t follows that a compressive mean stress w i l l i n -crease and a tensile mean stress w i l l decrease the fatigue strength* This assumes that the material i s not altered by the maximum stress applied whereby the fatigue strength would be influenced by structural changes* The fact that i n torsion the normal stress i s zero for a l l values of mean stress suggests that the mean stress i n torsion should have no effect i f the material i s not altered by the maximum stresses* As already mentioned, the predictions of this theory are consistent with the known trends of fatigue data* In 1959, F i n d l e y 1 1 re-examined and confirmed the theory to be acceptable i n explaining fatigue failure* The tests for this investigation were carried out on a Sonntag SF-l-U fatigue machine* This is a centrifugal force mechanical oscillator type machine with a capacity of 1000 lb* dynamic and 1000 lb* static load — o r a maximum load in one direction of 2000 lbs* Figure 1* shows a view of the Internal mechanism of this machine* The dynamic or vibratory force* generated by a mechanical oscillator and applied to a specimen or structure* is completely reversed and sinu-soidal* It is produced by an unbalanced mass (D). The oscillator shaft* through which the eccentric i s threaded* is driven by a synchronous motor at 1800 rpm through a flexible shaft assembly* The eccentric (B) i s threaded to enable adjustment of its unbalance to a required value which can be read directly in pounds of alternating force on the scale (G). The vertical component of the dynamic force is the only component transmitted to the specimen* The horizontal component is absorbed by four flexplates (B) which guide the oscillator assembly in the vertical direction* The fixed ends of the flexplates are held to the heavy welded frame which i s suspended from the cabinet by soft springs (E) to prevent transmission of vibrations to the floor* Two springs (A) which are fastened between the lower end of the oscillator and the frame are designed to absorb a l l inertia forces pro-duced by the vertical vibration of the oscillator housing and a l l other masses attached to and vibrating with i t * Thus* the dynamic force induced in the specimen i s equal to the eccentric setting, and remains so, irres-pective of the rigidity of the specimen or the amplitude of vibration* If the rigidity of the specimen changes during the test, then the amplitude - 10 -w i l l change too* thereby maintaining a constant repeated force i n the specimen* In order to function this way, springs (A) must be of such r i g i d i t y that when vibrating freely with the t o t a l reciprocating masses of the machine, the resulting natural frequency (without the specimen i n place) i s exactly equal to the operating frequency of 1800 cpm* The spring rate and frequency of operation are incapable of being altered and hence remain constant from test to test* Therefore, the to t a l equivalent reciprocating mass must be held constant to maintain a resonant condition* When the specimen i s mounted on the machine the system w i l l operate well below resonance* The weight which must be added to the empty vibrating cage to pro-duce a resonant condition i s known as the complementary weight and i s determined by calibration by the manufacturer, or i n the following manner: Set the eccentric for a small force of 2 to A lbs* Run the machine with various values of weights added to the vibrating cage, taking an amplitude reading each time* The correct complementary weight i s the one which pro-duces a maximum amplitude* At low values of weight the amplitude w i l l be small* I t w i l l build up to a higher value as the proper weight i s reached* Then further addition of weight w i l l bring the amplitude down again* A reset type of counter registers the number of repeated load cycles applied to the specimen* I t i s driven by a small synchronous motor which starts and stops automatically with the main motor* When a specimen fractures the main motor driving the unbalanced mass i s automatically turned off by one of the cut-off switches (F). The static or mean force i s applied to the specimen through two springs (A) which were mentioned earlier* Turning a screw (0) at the -11 -bottom of the machine causes the springs to either stretch or compress which then raises or lowers the o s c i l l a t o r assembly and so applies a pre-load to a specimen properly fastened i n a fixture* The springs are c a l i -brated and hence the static force i s measured by determining the deflection of the springs* A d i a l indicator (H) i s conveniently located to read this deflection* Figure 2. shows the fixture used i n the axial-load fatigue tests* I t has been shown by calibration tests performed by the manufacturer that this fixture produces less than 2 per cent extraneous bending stresses i n the specimen during mean tension tests. The torsional fatigue tests were performed using the fixture as shown i n Figure 3. Briefly, the alternating force produced by the re-ciprocating platen i s converted into an alternating torque i n the specimen by means of a crank arm. The crank arm i s pivoted i n two bearings at one end and oscillated on the other. The test specimen i s gripped between the o s c i l l a t i n g chuck and stationary chuck* A* Inertia Force Compensator Springs E. Suspension Springs B. Flexplates F. Cut-off Switches C. Scale G. Preload Screw Mechanism D. Eccentric H. Dial Indicator Figure 1* INTERNAL MECHANISM OF FATIGUE TESTING MACHINE Figure 2* FIXTURE FOR AXIAL LOAD FATIGUE TESTS Figure 3» FIXTURE FOR TORSIONAL FATIGUE TESTS 15 -% t e r & a l and Specimens The experiments reported herein were made on a mild steel supplied by Wilkinson Company Limited* This material was delivered i n the form of a 22-foot length of cold drawn seamless tubing l£ inches outside diameter by |-inch wall thickness* Its average composition was found to be: G, 0.10; Mn, 0*51; P, 0.013; S, 0.025; S i , 0*060. The original length of tubing was sawn transversely into 8-inch and 9-inch lengths. These lengths were then heated for 2 hours at 1200 deg. F. and a i r cooled to room temperature. This results i n the material being i n a homogeneous state throughout. A microscopic examination of several samples of the heat treated material was made and these indicate that this steel oan be regarded as clean. Nevertheless, longitudinal Inclusions are present and a typical example i s illustrated i n Figure A., which shows photomicrographs of polished and of etched sections. The inclusions are a l l longitudinally oriented because this i s the direction of cold drawing. The tensile fatigue specimens were made from the tube wall of the 8-inch lengths. Tube wall was used rather than solid bars because i t was convenient to use tubing for the torsion tests and the same material was thus used for the tension tests. Each 8-inch tube length was cut longitudinally into eight equal strips and these strips were machined into tensile test specimens. The dimensions of these are shown i n Figure 5« The specimens were finished by polishing with successively finer grades of emery paper and were then inspected to ensure that no serious scratches remained. Torsional fatigue specimens were machined from the 9-inch lengths. - 16 -The dimensions of these specimens are shown i n Figure 6. Since these are hollow specimens* special care was taken during machining to ensure concentricity* The specimens were finished the same way as was described with regard to the tensile ones* A requirement of torsional specimens when used i n the present machine i s that the specimen free length must be below a certain value* Calculations to show that this requirement was met are found i n Appendix A* a Polished x 720 b Polished and Etched x 720 Figure 4* 5H0T0MICROGRAPHS OF LONGITUDINAL SECTION OF A SPECIMEN 10" Bad. - POLISH SURFACE FINISH 0.155" Dla. t 0.005 If" 7f TURNED SECTION HOST BE CONCENTRIC WITH END SECTIONS Figure 5o AXIAL TENSILE FATIGUE SPECIMEN l£° Dta, MAKE FROM 3£ O.D. f I.D. SEAMLESS STEEL TUBING TURNED SECTION MOST BE CONCENTRIC WITH THE BORE Figure 6. TORSIONAL FATIGUE SPECIMEN Test Program A x i a l - l o a d endurance t e s t s were performed at four values o f t e n s i l e mean s t r e s s . These mean stresses were zero. 10,000 p s i , 20,000 p s i , and 30,000 p s i . The t e s t s i n t o r s i o n a l fatigue were c a r r i e d out at mean stresses o f zero, 5,000 p s i , 10,000 p s i , and 15,000 p s i . Ah endurance l i m i t based on 10 m i l l i o n cycles of s t r e s s was established f o r each mean s t r e s s . The procedure o f t e s t i n g was the same f o r both the a x i a l - l o a d and t o r s i o n a l f a t i g u e t e s t s . At each new value o f mean s t r e s s , a specimen was u s u a l l y f i r s t tested a t a low a l t e r n a t i n g s t r e s s . I f i t withstood 10^ cycles, the a l t e r n a t i n g s t r e s s was r a i s e d and the specimen was r e -7 tes t e d . I f i t withstood another 10 cycles, then the a l t e r n a t i n g s t r e s s was r a i s e d again and the procedure repeated u n t i l the specimen f r a c t u r e d . The o r i g i n a l run and the r e t e s t runs were then a l l p l o t t e d on a S-N diagram. Since a rough estimate o f the endurance l i m i t f o r the given mean s t r e s s was obtained from the t e s t runs made on the f i r s t specimen, the remaining specimens were tested a t stresses which would give a curve w e l l defined i n the mortal and endurance regions. In the case o f a x i a l - l o a d t e s t s , the required f o r c e , e i t h e r a l t e r -nating or mean, was determined by m u l t i p l y i n g the given s t r e s s by the c r i t i c a l c r o s s - s e c t i o n area o f the specimen. For the t o r s i o n a l t e s t s , the required f o r c e , e i t h e r a l t e r n a t i n g or mean, was c a l c u l a t e d by means of the t o r s i o n formula P • S 8j/Rc i n which S s i s the given shearing s t r e s s , J i s the p o l a r moment o f i n e r t i a o f the cros s - s e c t i o n where the c r i t i c a l s t r e s s occurs, B i s the crank arm, and c i s the outside radius o f the c r i t i c a l c r o s s - s e c t i o n . 21 -The static and dynamic stresses on the specimen according to the machine* both for an axial-load and torsional test* were checked using e l e c t r i c a l resistance strain gauges. The details of this verification are given i n Appendix B. I t has been noted that i n some of the hollow specimens used the bore was s l i g h t l y eccentric i n relation to the outer diameter. A cor-rection i n the stress estimate, which has been applied to allow for the stress concentration due to eccentricity, i s based on the constant shear flow theory. This i s an approximation which states that i n a thin tube subjected to torsion the product stress x wall thickness i s constant* Essentially* the above correction i s based on the equilibrium between the moment of stresses within a specimen and the external torque; i t i s therefore applicable* provided that no la t e r a l distortion of a specimen occurs* i n the plastic as well as i n the elastic range. I t may be added that the fatigue fracture i n a l l the hollow specimens occurred at the minimum wall thickness. The corrections applied to the stress have been smaller than 5 per cent i n a l l cases* In a l l torsional tests with a superimposed static stress i t has been observed that the mean stress of a cycle decreases even though the maximum stress of the cycle does not exceed the static yield stress. This may probably be due to slippage i n the grips. Consequently, i t has been necessary to adjust the mean stress at intervals depending upon the applied range of stress. For example, i n tests below the fatigue l i m i t a small adjustment has been required only i n the early part of a test; on the other hand, i n some tests at higher ranges of stress i t has been necessary to adjust the mean stress every hour or so i n order to maintain i t within 2 per cent of the required value. 22 -Test Results The results of this investigation are divided into two main sections* static and fatigue* The static results are presented quantitatively while the fatigue results are a l l presented i n graphical forme Static* Most of the static tensile tests were carried out i n the Instron tensile testing machine* This machine incorporates a highly sen-s i t i v e electronic weighing system employing bonded-wire strain gauges for detecting and recording the tensile load applied to the specimen under test* The moving crosahead, to which the lower pulling jaw i s attached* i s operated by two v e r t i c a l drive screws from a unique positional-servo drive that provides a considerable f l e x i b i l i t y of control over the motion of this jaw* The upper pulling jaw i s connected to the load c e l l by means of a flexible coupling to provide self-alignment with the specimen* Thus this machine has the virtue that i t delivers a concentric loading with no extraneous bending stresses incurred at a l l * Another important feature i s that the load weighing system i t s e l f exhibits essentially no mechanical inertia* Therefore* i t does not influence through i t s own action the properties of the specimen to be measured* In addition to these tests i n the Instron machine* a few subsidiary tests to break were carried out i n a Baldwin hydraulic machine* Results of the static tensile tests of the subject material show same variation between individual tests* The averages of the available data as obtained by the Instron machine indicate: an ultimate strength of 54*600 p s i , an upper yield point of 53,800 psi, a lower yi e l d point of 41,100 p s i , and an elastic modulus (extensometer employed) of 31*6 x 10^ p s i * The tests done In the Baldwin hydraulic machine give a lower y i e l d - 23 -point of 39.200 ps i and an ultimate strength of 53,300 p s i . Appendix C gives a tabular account of the static tensile tests. The upper y i e l d point i n torsion was found by assuming the material used to follow the maximum energy of distortion theory i n yielding. For an isotropic material, the maximum energy of distortion theory predicts yielding when the shear stress i s equal to 0.58 x yield stress i n tension. This i s an accurate prediction of the shear stress at which yielding would occur i n a bar of ductile material stressed i n torsion. Using this re-lationship (the photomicrographs show almost no preferred orientation of crystals which means that the material i s nearly isotropic), the upper yield stress i n torsion was found to be 31,000 p s i . This i s the only static torsional property required for this thesis. Fatigue. The presentation of results i s simplified by the fact that under a l l of the stress conditions investigated no macroscopic plastic flow precedes rupture by fatigue. The stresses quoted are thus a l l actual stresses© Appendix D gives a tabular account of the fatigue test data. Figure 7. shows the S-N curves obtained from the axial-load tests with tensile mean stresses zero, 10,000 p s i , 20,000 p s i , and 30,000 p s i . The estimated fatigue limits are ± 28,500 p s i , ±27,800 p s i , ± 25,800 p s i , and ± 23,500 ps i respectively. A summary of the effect of the mean stress on the fatigue strength i s ill u s t r a t e d i n Figure 8., where the endurance limits are shown as a function of the mean stress. Details of the torsional tests are presented i n Figure 9» Examin-ation of these S-N curves gives estimated fatigue limits of ±16,600 p s i , ± 16,800 p s i , ± 16,500 p s i , and ±16,500 psi for mean stresses zero, 5,000 psi, 10,000 psi, and 15,000 p s i respectively. In Figure 10. a ratio - 24 -f a t i g u e l i m i t a t a given mean s t r e s s / f a t i g u e l i m i t a t zero mean s t r e s s i s p l o t t e d against a r a t i o maximum shearing s t r e s s o f range/shearing tipper y i e l d strength* This i l l u s t r a t e s the e f f e c t of the mean s t r e s s on the f a t i g u e strength i n shear* I n determining the endurance l i m i t s f o r the various mean stresses from the a x i a l - l o a d and t o r s i o n a l f a t i gue t e s t s , i t i s to be pointed out that most o f the emphasis vas placed on points representing o r i g i n a l t e s t runs rather than on r e t e s t p o i n t s * However, the e f f e c t of under-s t r e s s i n g i s seen to be appreciably small* being n e g l i g i b l e i n the t o r s i o n a l t e s t s * I f the r e t e s t points were taken i n t o account i n the a x i a l - l o a d t e s t s , the slope o f the mortal region o f the S-N curve would change a small amount, but the endurance region would be unaltered so that the f a t i g u e strength would remain the same* - 25 -a NUMBER OF CYCLES TO FAILURE Figure 7. AXIAL FATIGUE TESTS WITH VARIOUS MEAN STRESSES a Reversed stress* b Mean stress +10,000 lb per sq* in* o Mean stress +20,000 lb per Bq. in* d Mean stress 4-30,000 lb per sq* in* o+- Unbroken specimen* • Re tested unbroken specimen* - 26 -\ \ \ \ \ \ V \ \\ \ \ ^ \ 0 12,000 24,000 36,000 48,000 60,000 MEAN STRESS, PSI Figaro 8. EFFECT OF TENSILE MEAN STRESS OH THE FATIGUE STRENGTH UNDER AXIAL LOAD Modified Goodman Lav Sodar berg Line Gerber Parabolic Relationship NUMBER OF CYCLES TO FAILURE Figaro 9* TORSIONAL FATIGUE TESTS WITH VARIOUS KSAS STRESSES a Reversed stress* b Msan stress 5,000 lb per eq* in* o Mean stress 10,000 lb per sq* in* d Mean stress 15*000 lb per sq* in* o* Unbroken speelmeno • Retested unbroken speoimen* - 28 -1.1 1.0 0.9 0.8 0.7 o 0 o ^ ^ ^ ^ 0 0.2 0.4 0.6 0.8 1.0 J&XDBM SHEARING STRESS OF BANGS SHEARING UPPER HELD STRENGTH Figaro 10. EFFECT OF MEAN STRESS 05 THE FATIGUE STRENGTH IN SHEAR DUE TO TORSION Present Re salts - e — e — e -Snith — Chodorovski - 29 -Cn - r ra-l^np «rtf M a n s i o n of Test Results The dependence of the yield point and ultimate strength i n low-carbon steel on the rate of straining made i t d i f f i c u l t to correlate with certainty the mean static properties with the behavior i n fatigue* I t seemed advisable and less misleading to give the overall variations i n properties found* As mentioned before* this data i s found i n Appendix C. In one particular test of the axial-load series, the maximum stress of the cycle was 55.000 p s i * The measured average upper yield stress was found to be 53,800 psi which indicates that permanent deformation of the specimen should have resulted* However, no yielding or measurable de-formation was noticed at a l l * This can be explained by the fact that the rate of straining of the fatigue machine i s very high: there i s reason to suppose that the value of the upper y i e l d stress at this high rate of strain was above the maximum stress of the cycle* For this particular material the measured upper y i e l d stress was nearly that of the ultimate strength* This i s at f i r s t sight surprising* One i s perhaps suf f i c i e n t l y accustomed to the value of upper yield stress as being 50 to 60 per cent of the ultimate strength* Yet the clue to this phenomenon may perhaps be found i n the two important characteristics of the Instron testing machine: concentricity of loading and low mechanical i n e r t i a of the loading mechanism* Both these characteristics have been referred to earlier* I t may be postulated that this causes yielding to occur uniformly throughout the entire cross-section rather than i n l o c a l -ized sections as would be the result of a loading that i s eccentric* Since localized yielding would be accompanied by a redistribution of stress, i t - 30 -follows that an eccentric loading gives a value of upper yield stress which may be much lower than obtained by a concentric loading. The tests done i n the Baldwin hydraulic machine seem to bear out this fact* The s l i g h t l y eccentric loading of this machine gave results appreciably lower than those obtained l n the Instron machine* Mild steel was used exclusively for this investigation* but this does not r e s t r i c t the interpretation of fatigue results to nfilQ steel only* The information gained from this study can be applied to most ductile ferrous materials* Figure 8* shows the effect of tensile mean stress on the fatigue strength under axial-load* The points representing the endurance limits are joined by a solid line* and there i s no evident reason for drawing this curve other than that shown* At the same time curves expressing the Goodman* Soderberg, and Gerber relationships for the effect of tensile mean stress on the fatigue strength are shown* I t i s clear from this i l l u s t r a t i o n that a l l three relationships are conservative i n determining the effect of the mean stress* The Gerber parabolic relationship, how-ever, f i t s the present data most closely* In comparing results from other sources these s t r i c t comparisons can seldom be made, and i t must be expected that close agreement cannot always be obtained* Discrepancies can arise as a result of variations In the properties of the same materials, the method of selecting samples* and differences i n testing technique* Previous investigations (Smith, Findley, Morrison and O'Connor, Sines, and others) have shown that the fatigue strength decreases with increasing tensile mean stress: the results of this investigation show - 31 -the same* I t should be pointed out* though, that e a r l i e r efforts nearly a l l show a linear relation between the mean stress and fatigue strength while the present relation definitely follows a parabolic trend* Even i f the accuracy i n the determination of the endurance limits, which i s within ± 500 psi* was to be taken into account, a parabolic curve would s t i l l f i t the data more closely than a straight line* No experiments were performed here to determine the behavior of the fatigue strength under axial-load with compressive mean stress* But i t has been found by others (Findley, Morrison and O'Connor, and Sines) that an increasing compressive mean stress causes the fatigue strength to i n -crease and hence i t seems safe to state that the same effect would have been observed for this steel had tests been carried out* Reference to Figure 10* shows the effect of mean stress on the fatigue strength i n shear due to torsion* Had a few more specimens been available i t might have been possible to reduce the range of uncertainty i n the values obtained* As i t i s there would seem to be no j u s t i f i c a t i o n for joining the experimental values by anything other than a straight l i n e * Comparison of torsional fatigue data with the results of other investigators indicates very good agreement with Smith"', but disagreement with Chodorowski^* Smith showed that the mean stress has no effect on the fatigue strength as long as the maximum stress of the cycle did not exceed the shearing yi e l d strength* The data presented i n Figure 10. i s i n complete agreement with this* Chodorowski, however, concluded that the fatigue strength decreased linearly with increase i n mean stress* This i s certainly contrary to the findings made here* N© positive explanation of this rather appalling discrepancy can be suggested, but -32 -i t i s f e l t that differences i n the ratio of the wall-thickness to the external-diameter may have contributed to this variance. In the tests conducted by Ghodorowski the ratio was l/26 while i n the present tests the ratio was l/8. I t i s worth while now to examine i n the li g h t of the results of this investigation the various theories that have been put forth i n ex-plaining fatigue f a i l u r e . Reviewing b r i e f l y , i t was considered by F i n d l e y 1 1 that the i n -ception of fatigue resulted from alternating shearing stresses and that the resistance to fatigue was influenced by the magnitude and sign of the steady normal stress on the plane of maximum shear stress. The resistance to fatigue would be increased by compressive values and decreased by tensile values of steady normal stress. Since the steady normal stress i s directly affected by the mean stress, a compressive mean stress w i l l increase and a tensile mean stress w i l l decrease the fatigue strength. In torsion, the fact that the steady normal stress i s zero on the plane of maximum shear stress for a l l values of mean stress suggests that the mean stress i n torsion should have no effect on the fatigue strength i f the material i s not altered by the maximum stresses. I t i s readily seen from Figures 8. and 10. that this theory by Findley accurately explains the damage done to the material structure. In accord-ance with the theory, the tensile mean stresses operated directly to i n -crease the l i a b i l i t y to failure whereas i n torsion the mean stress had no effect on f a i l u r e . As i t happens, tests with compressive mean stress were not performed, but i t has been stated i n ea r l i e r sections of this thesis that certain investigators, for example, Morrison and G^Connor1, have 33 -shown that the fatigue strength increases with compressive mean stress* This again verifies the theory of failure as proposed by Findley* For the case of axial-load tests, no d i f f i c u l t y arises i n disproving the theory of fatigue failure as proposed by Orowan-^ and FrendenthaT^* This theory suggested that the fatigue strength was independent of the mean stress as long as the elastic l i m i t was not reached* None of the tests made i n this study had the maximum stress of the cycle above the upper yield point: yet the fatigue strength was affected by even the smallest tensile mean stress* I t follows that the fatigue strength i s definitely dependent on the mean stress* I t has also been suggested that structural changes produced by the maximum stress of the cycle influence the fatigue strength rather than the mean stress i t s e l f * Again i t i s to be pointed out that the stresses employed were below the upper yield point and thus the effective structure of the material was not altered* Since this eliminates the structural effect, the mean stress remains as the only important factor i n the expla-nation of the behavior of fatigue failure* In passing, attention i s drawn to the statement made earli e r that the material was assumed to follow the maximum energy of distortion theory i n yielding* This theory predicted that the ratio y i e l d stress i n torsion/ yi e l d stress i n tension was equal to 0.58: i f this i s true, then the ratio fatigue strength i n shear at zero mean stress/fatigue strength under axial-load at zero mean stress should also be equal to 0.58. As i t i s , exami-nation of the data shows this ratio to be 16,600/28,500 or exactly 0.58 which j u s t i f i e s the assumption made* - 34 -Conclusiona and Recommendations Conclusions. From the interpretation of the data which have been presented, the following conclusions are drawn: 1* The fatigue strength under axial-load of the steel used i s affected by even a small tensile mean stress; i n the region investigated* which extends from reversed to repeated stresses, the endurance l i m i t decreased with the increase i n mean stress* 2* The mean stress has no effect on the fatigue strength i n shear provided that the maximum stress of the cycle does not exceed the torsional static shearing yield strength* 3* The fatigue strength i s not Influenced by any structural changes i f the maximum stress of the cycle i s below the elastic l i m i t * Um The normal stress acting on the plane of maximum shear stress was; found to be an important factor i n fatigue failure* Bfloo^ftpdations* The following recommendations are made as future Investigations: 1* Axial-load fatigue tests with compressive mean stresses using specially designed specimens that would minimize the occurrence of bending stresses and buckling© 2* Static tensile tests with varying degrees of eccentricity of load-ing to determine the effect on the properties of a material* - 35 -APPENDIX A Calculation of Torsion^, S p ^ c ^ n Free Length The maximum specimen free length permissible i s given by the formula 2E3 S where L * specimen free length, inches I = | amplitude of platen, inches (maximum) s 0.37 Q s dynamic modulus of r i g i d i t y , p s i (shear modulus) s 12 x 10^ D = smallest specimen diameter, inches • 1.0 R s crank arm. inches s 15.25 S Q : alternating shearing stress, p s i (maximum used) s 20,000 L s Qi37 x 13 as IO6 x. I.O 2 x 15.25 x 20,000 = 7.27 I t i s recommended that the free length be 75 per cent or less of the length calculated above i n order to allow automatic adjustment of amplitude with change of dynamic modulus during the fatigue test. The recommended free length of the torsional specimens i s thus 0.75 x 7.27 8 5*45 inches or les s . The free length used i n the torsional tests was 3.63 inches. Therefore, the requirement of specimen free length was f u l l y s a t i s f i e d . - 36 -APPENDIX B Verification of Machine Loads Axial-load Test: Static - Tensile stress on specimen according to the machine S 30,000 p s i . - Strain by e l e c t r i c a l resistance strain gauges - 1000 micro inches© - Stress by strain gauge i s given by S s strain x E s 1000 x IO" 6 x 30 x 10 6 s 30,000 p s i . Dynamic - Maximum tensile stress on specimen according to the machine • 30,000 p s i . - Strain by e l e c t r i c a l resistance strain gauge (oscilliscope used) = 1000 micro inches. - Stress by strain gauge i s given by S = strain x E * 1000 x 10" 6 x 30 x 10 6 « 30,000 p s i . Therefore, both the static and dynamic loadings of the machine give correct values of stress. Torsional Tests: Static - Shear stress on specimen according to the machine « 15,000 p s i . - Strain by e l e c t r i c a l resistance strain gauge s 640 micro inches. - Shear stress by strain gauge i s given by S 8 = p t r a i a s B - 640 1 K T 6 a 3Q x. 3.Q6 = 14,775 p s i . 1.3 1.3 - 37 Dynamic - Maximum shear stress on specimen according to the machine • 13,850 p s i . - Strain by e l e c t r i c a l resistance strain gauges (oscilHscope used) s 602 micro inches* • Shear stress by strain gauge i s given by S s - afrgXn x E - 602 x 1Q-& x 30 x 10* - 13,900 p s i . 1*3 1.3 The error involved here i s less than 2 per cent i n both instances and so can be neglected* Therefore, as i n the case of the axial-load test, the static and dynamic loadings of the machine give correct values of stress* This indicates that the machine calibration i s s t i l l i n order* 38 -APPENDIX C Statie Tensile Tests Tests performed i n the Insiron testing machine: Bate of straining inches/minute Upper yi e l d stress, p s i Lower yi e l d Ultimate tensile stress, p s i strength, psi 0.05 56,100 53,600 50,500 47,000 40,000 40,400 41,700 40,100 52,200 53,300 54,500 52,000 0.10 55,300 53,500 56,600 54,900 41,100 41,600 42,200 41,500 54,500 54,600 54,300 55,100 5.0 53,000 57,000 10.0 56,500 55,000 56,400 57,000 Average 53,800 41,100 54,600 Tests performed i n the Baldwin hydraulic machine: Lower yield point, p s i Ultimate tensile strength, p s i 39,400 38,900 52,600 53,900 Average 39,200 53,300 - 39 -APPENDIX D F a t i g u e T e s t D a t a Axial-load fatigue tests with various tensile mean stresses: Specimen Alternating Stress Cycles Psi Mean Stress Psi 10,000 20,000 30,000 2 3 4 2 3 A 2 3 A 2 3 4 27,000 28,000 29,000 29,000 26,000 27,000 28,000 29,000 28,000 27,000 28,000 24,000 25,000 26,000 27,000 26,000 25,000 26,000 22,000 23,000 24,000 25,000 24,000 23,000 24,000 7 10L unbroken 10 Unbroken 3,354,000 3,702,000 1,703,000-30,000 1,703,000-28,000 10' Unbroken 29,000 2,780,000 10' 10? 4*960,000 1,062,000 7,032,000-10' 2,370,000 10? 10 ' 5,430,000 1,283,000 6,147,000-10 ' 3,922,000 10 7 5,710,000 1,231,000 4,437,000, Unbroken Unbroken Unbroken Unbroken Unbroken Unbroken Unbroken Unbroken 10' Unbroken 4,880,000 40 Torsional fatigue tests with various mean stresses: Mean Stress Psi Specimen Alternating Stress Psi Cycles 0 1 2 3 4 17,400 16,000 17,000 18,000 15,000 17,000 4,671,000 10' Unbroken 6,009,000 2,695,000 10' Unbroken 6,735,000 5,000 1 2 3 4 17,900 16,000 17,100 17,300 18,400 17,600 870,000 10' Unbroken 3,453,000 , 10' Unbroken 638*000 2,501,000 10,000 1 2 3 15,300 16,300 17,400 17,300 16*700 10^ Unbroken 10' Unbroken 1,220,000 2,040,000 4,248,000 15,000 1 2 3 16,100 17,100 18,600 17,400 10^ Unbroken 4,727,000 737,000 2,076,000 - 41 BIBLIOGRAPHY 1. Morrison, J * L. M., and H. C. O'Connor* The E f f e c t of Ma an Stress on the Push-Pull g a t i g u e Properties of an A l l o y S t e e l . I n t e r n a t i o n a l Conference on Fatigue o f M9tals, 1956. 2* Woodward, A, R., K. W. Gunn, and G. F o r r e s t . The. E f f e c t a£ M9an  Stress aa tfee. Fatigue o f Aluminium A l l o y s . I n t e r n a t i o n a l Conference on Fatigue o f Metals, 1956. 3* Gough, H. J . Fatigue o£ Metals., New York: D. Van Nostrand, 1926. 4. Soderberg, C. R. "Factors o f Safety and Working Stresses," Transactions. Am. SQC.. Mechanical Engrs.. Applied Mechanics D i v i s i o n , V o l . 52, 1930, p. 13; "Working Stresses," Transactions. Ag. Soc. Mftt».h«Tvf n ^ l , Eners.. Applied Mechanics D i v i s i o n , V o l . 55, 1933, p. 131. 5. Smith, J . 0. The E f f e c t g£ Range g£ S t r e s s o£ J2l& IjMEUZ Strength, fl£ Matals. U n i v e r s i t y o f I l l i n o i s Engineering Experiment S t a t i o n B u l l e t i n No. 334, 1942. 6. Findley, W. N., F. G. Mergen, and A. H e Rosenberg. "The E f f e c t o f Range o f S t r e s s on Fatigue Strength o f Notched and Unnotched SAE 4340 S t e e l i n Bending and Torsion," Proceedings. Am. Sbc. Testing Mats*. V o l . 53, 1953, p. 768. -7. F i n d l e y , W. N. "Experiments i n Fatigue Under Ranges o f S t r e s s i n Torsion and A x i a l load From Tension to Extreme Compression," Proceedings^ Am. So£. Testing Mats*. V o l . 54, 1954, P* 836. 8. Sines, G«, and J . L. Waisman* Metal Fatigue. U n i v e r s i t y o f C a l i f o r n i a Engineering Extension S e r i e s * New York: McGraw-H i l l Book Co., Inc., 1959* 9* Smith, J . 0. Tfee. E f f e c t o£ Ranee o f Stress on. ihe. Torsipnajl, Fatjffle Strength a£ S t e e l * U n i v e r s i t y of I l l i n o i s Engineering Experiment S t a t i o n B u l l e t i n No. 316, 1939* 10. ChodorowskL, W. T. Fatigue Strength jLg Shear o l m A l l o y S t e e l With P ^ , ^ i f f l r Reference j& the E f f e c t o f Mean, St r e s s and, D i r e c t i o n a l P r o p e r t i e s . I n t e r n a t i o n a l Conference on Fatigue o f Metals, 1956. 11. Findley, W. N. "A Theory f o r the E f f e c t o f Mean Stress on Fatigue of Metals Under Combined Torsion and A x i a l Load or Bending," Transactions. Am. Spg. Mechanical Eners.. S e r i e s B, November, 1959, p. 301. 12. Orowan, E. "Theory o f the Fatigue o f Mstals," Proceedings. Royal SQC* (London), Se r i e s A, V o l . 171, 1939, p. 79* 42 Freudenthal, A. M. "The Statistical Aspect of Fatigue of Materials," Proceedings. Royal Soca (London), Series A, Vol. 187, 1946, p. 416. 

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