Brigham Young UniversityBYU ScholarsArchive

All Theses and Dissertations

1965-6

Residual Stresses and the Bauschinger EffectRaymond M. HuebnerBrigham Young University - Provo

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BYU ScholarsArchive CitationHuebner, Raymond M., "Residual Stresses and the Bauschinger Effect" (1965). All Theses and Dissertations. 7122.https://scholarsarchive.byu.edu/etd/7122

RESIDUAL STRESSES AND THE

BAUSCHINGER EFFECT

A Thesis

Subm itted to th e M echanical E ng ineering Department

Brigham Young U n iv e rs ity

Provo, Utah

In P a r t i a l F u lf i l lm e n t

Of th e Requirem ents fo r th e Degree o f

M aster o f Science

t>y

Raymond M. Huebner

June, 1965

This t h e s i s , by Raymond M. Huebner, i s accep ted in i t s p re sen t

form by th e Department o f M echanical E ng ineering o f Brigham Young

U n iv e rs ity as s a t i s f y in g th e th e s i s requ irem en ts f o r th e degree o f

M aster o f S c ience .

Date

Typed by fcfery S. Thornton

ACKNOWLEDGEMENT

The a u th o r w ishes to exp ress h is a p p re c ia tio n to the members o f h is

a d v iso ry committee and t o th e Department o f M echanical E ng ineering a t

Brigham Young U n iv e rs ity . P a r t i c u la r a p p re c ia tio n i s extended to D r.

John M. Simonsen and C harles M. P e rc iv a l fo r t h e i r h e lp and guidance.

This th e s i s i s d ed ica te d to th e a u th o r 's p a re n ts , Mr. and Mrs.

Raymond H. Huebner fo r t h e i r con tinued encouragement th roughou t h is

e d u ca tio n .

TABLE OF CONTENTS

iv

C hapter Page

I . INTRODUCTION ............................................................................................. 1

I I . LITERATURE SURVEY.....................................................................................5

I I I . METHOD OF APPROACH.....................................................................................9

Equipment Used Procedure Data R eduction

IV. DISCUSSION OF THE MEASUREMENTS AND RESULTS............................. 3*+

E f fe c t o f E ccen tr ic ity -C a lc u la tio n o f th e Axis o f Average S tra in I n te r p r e ta t io n o f th e MeasurementsI n te r p r e ta t io n o f th e R esu lts

V. CONCLUSIONS AND RECOMMENDATIONS.............................. k j

BIBLIOGRAPHY............................................................................................................U8

APPENDIX 50

CHAPTER I

INTRODUCTION

W ithin th e l a s t te n years th e re has been much i n t e r e s t in th e

f i e l d o f e l a s to - p l a s t i c b ehav io r o f en g in ee rin g m a te r ia ls . Hie re c e n t

p o p u la r i ty o f th e lo w -co st l ig h t-w e ig h t p l a s t i c a l l y designed s t r u c ­

tu re s h as added a g re a t d e a l o f im petus to t h i s i n t e r e s t . This th e s i s

d e a ls w ith one o f th e a sp e c ts o f e l a s to - p l a s t i c b eh av io r, namely, th e

d i s t r ib u t io n o f th e r e s id u a l s t r e s s e s in a sim ple member which has been

p l a s t i c a l l y deformed in te n s io n a n d /o r compression,. In p a r t i c u l a r , an

a ttem p t has been made to r e l a t e th e se r e s id u a l s t r e s s e s to th e w e ll-

known b u t l i t t l e understood B auschinger e f f e c t .

The B auschinger e f f e c t i s a lo s s o f y ie ld s tre n g th in one d i r e c ­

t io n which r e s u l t s from a member's be ing p re v io u s ly y ie ld e d in th e

o p p o site d i r e c t io n . For example, when a member i s p l a s t i c a l l y s t r e s s e d

in te n s io n , th e B auschinger e f f e c t causes th e y ie ld p o in t in com pression

to occur a t an a p p re c ia b ly low er s t r e s s th an would be exp ec ted . This

i s shown in F ig u re 1 by th e low er com pressive y ie ld s t r e s s which occurs

a t p o in t D. The occurence o f t e n s i l e y ie ld a ls o causes a n o th e r phenom­

enon, c a l le d work h a rd en in g , which causes th e n ex t y ie ld p o in t to occur

a t a s t r e s s h ig h e r th an th e f i r s t one. A lo g ic a l ex p lan a tio n f o r t h i s

beh av io r would be th a t r e s id u a l s t r e s s e s a re formed in th e member in

such a way as to a id th e t e n s i l e y ie ld s tre n g th in one sense and to

h in d e r th e com pressive y ie ld s tre n g th in th e o th e r .

1

2

I t i s p o ss ib le th a t work harden ing may s tre n g th e n th e m a te r ia l

to th e p o in t where th e com pressive y ie ld p o in t o f th e s t r a in e d m a te r ia l

occurs a t a h ig h e r s t r e s s th an th e com pressive y ie ld p o in t o f th e o r ig ­

in a l u n s tra in e d m a te r ia l . In F ig u re 1, th e l in e GH i s shown lo n g er than

OE.T jV -l

The Bauschinger e f f e c t s t i l l e x is t s because th e r a t i o ^ - equals BCthe r a t i o ——- .CD

---------- Sm all s t r a in---------- Large s t r a in

F ig . 1 . --M echanical H y s te re s is

When a b a r i s a x ia l ly loaded in a t e n s i l e t e s t in g m achine, the

h ig h e s t s t r e s s e s a re known to occur in th e outerm ost f i b e r s . I t i s

3

q u ite n a tu ra l t o assume th a t th e se f ib e r s a re th e f i r s t to y ie ld and

un load . When th e e x te rn a l lo ad on th e b a r i s re la x e d , th e se le s s e r - lo a d e d

o u ts id e f ib e r s w i l l th en be pushed in to com pression by th e c o n tra c tio n o f

th e o th e r f i b e r s . Of co u rse , th e com pressive s t r e s s e s o f th e o u ts id e

f ib e r s would be balanced by t e n s i le s t r e s s e s in th e o th e r f i b e r s . A

rough sk e tch o f t h i s expected r e s id u a l s t r e s s d i s t r ib u t io n i s shown in

F ig u re 2 .

Tension

F ig . 2 . --E xpected r e s id u a l s t r e s s d i s t r ib u t io n in a p l a s t i c a l l y deformed t e n s i l e specim en.

An a ttem p t was made in t h i s s tu d y to show th a t t h i s s i tu a t io n

does occur and th a t i t i s th e cause o f th e B auschinger e f f e c t . This was

accom plished by m easuring and p lo t t in g th e r e s id u a l s t r e s s p a t te rn s

found in b a rs o f s t e e l which had been p re v io u s ly p l a s t i c a l l y deform ed.

The method o f beam d is s e c t io n was s e le c te d f o r th e measurement o f th e se

s t r e s s e s .

k

The in f lu e n c e o f r e s id u a l s t r e s s e s upon th e m echanical p ro p e r t ie s

o f en g in ee rin g m a te r ia ls has been re c e iv in g a c c e le r a te d a t t e n t io n w ith in

th e p a s t two decades. Knowledge in t h i s f i e l d has been r e s t r i c t e d by

th e g e n e ra l d i f f i c u l t y in m easuring th e se r e s id u a l s t r e s s e s . To d a te

th e on ly common methods in use a re ( l ) th e d is s e c t io n method, where a

member i s cu t a p a r t and th e r e s id u a l s t r e s s e s a re c a lc u la te d from th e

r e s u l t in g geom etric changes in th e p a re n t member, and (2) th e X -ray

d i f f r a c t io n method.

The X -ray d i f f r a c t io n method i s a p p a re n tly th e one most commonly

u sed . I t ' s p o p u la r ity i s p robab ly due to th e speed and ease w ith which

measurements can be made. The techn ique r e l a t e s th e d i f f r a c t io n o f an

X -ray beam to th e d i s to r te d l a t t i c e s t ru c tu re in m eta ls caused by r e s i ­

d u a l s t r e s s e s . Experience has shown th i s m iso r ie n ta tio n to be g re a te s t

a long th e a x is o f maximum sh e a r . As would be expected , on ly th e s t r e s s e s

on th e su rfa ce o f th e specimen a re m easured.

This n o n -d e s tru c tiv e method has obvious a d v an tag es . I f the

s t r e s s e s on th e c ro ss s e c tio n o r o th e r in te r n a l lo c a t io n s a re d e s ire d ,

th e specimen must be c a r e f u l ly sawed and p o lish e d .

In th e beam d is s e c t io n method of r e s id u a l s t r e s s measurement, th e

a x ia l s t r e s s e s a t any p o in t in th e specimen may be m easured. This i s

accom plished by c u t t in g s le n d e r beams out o f th e b a r and by m easuring

th e change in d e f le c t io n and hence c u rv a tu re a s la y e rs to g e th e r w ith

t h e i r co n ta in ed s t r e s s e s a re removed. This removal i s done by g rin d in g

w ith ex trem ely l i g h t c u ts . The S .A .E . p u b lic a tio n TR-1^7 ( l ) g ives d e ­

t a i l s and an e v a lu a tio n o f th e se and o th e r m ethods. The techn ique used

a t th e G eneral Motors L ab o ra to rie s fo r th e d is s e c t io n method (2) has been

in c lu d ed in Appendix A. This was done fo r th e convenience o f th e re a d e r

s in ce t h i s document i s n o t r e a d i ly a v a i la b le .

CHAPTER I I

LITERATURE SURVEY

A l i t e r a t u r e search was made to determ ine what work had a lre a d y

been perform ed in th e f i e l d o f r e s id u a l s t r e s s e s . In p a r t i c u la r , i n ­

fo rm ation l in k in g r e s id u a l s t r e s s e s to th e B auschinger e f f e c t was sough t.

Many papers were found which d e a l t w ith th e s u b je c t o f r e s id u a l

s t r e s s e s . In a lik e w ise manner, th e B auschinger e f f e c t was found to be

a popu lar to p i c .

S . L. Smith and W. A. Wood (3 ) have re p o rte d f in d in g p l a s t i c a l l y

induced r e s id u a l su rfa ce s t r e s s e s which were o r ie n te d in such a manner

to oppose the fo rm erly a p p lie d lo a d . These su rfa ce s t r e s s e s were found

to be balanced by op p o site s t r e s s e s n e a r th e c e n te r . T h e ir measurement

o f th e se su rfa ce s t r e s s e s was done by X -ray d i f f r a c t io n m ethods. E tch ­

in g a c ro ss th e c ro ss s e c tio n showed th e expected l a t e r a l d isp lacem ents

in th e c r y s ta l l a t t i c e .

Much o f th e ex p erim en ta tion in t h i s f i e l d has been done w ith

to r s io n o f tu b u la r samples o f aluminum, b ra s s , copper, i r o n , le a d and

n ic k e l . A permanent deform ation c a l le d B auschinger s t r a i n has been ob­

served in some m e ta ls . Some a ttem p ts have been made to c o r r e la te t h i s

s t r a in w ith r e s id u a l s t r e s s e s . R. L. Wooley (h) has r e la te d th e Bausch­

in g e r e f f e c t to s l i p a s o th e r au th o rs have a ls o done. This same a r t i c l e

a ls o d isc u sse s methods o f p re d ic t in g th e amount o f th e B auschinger

s t r a i n . Wooley d e fin e s th e B auschinger s t r a in as th e d if fe re n c e in

s t r a in in com pression between th e ( p l a s t i c a l l y in te n s io n ) deformed and

5

6

undeformed specim ens. Naghdi, Essenburg and Koff (5 ) have t r i e d combin­

a t io n s o f to r s io n and te n s io n , and th e r e s u l t i s a " y ie ld s u r fa c e ” on the

s t r e s s - s t r a i n diagram . In t h i s c a se , a B auschinger e f f e c t from to r s io n

b u t n o t te n s io n was observed .

The g ra d u a lly in c re a s in g y ie ld p o in ts in te n s io n r e s u l t in g from

su ccess iv e cy c le s o f p la s t i c deform ation o r work h a rd en in g , a s i t i s

commonly c a l le d , has in s p ire d many th e o r ie s o f y ie ld s t r e n g th . C ry s ta l­

l in e a c t io n has re c e iv e d much a t t e n t io n . A. H. C o t t r e l l (6) has

th e o r iz e d a d i s t r ib u t io n o f th e s t r e s s e s among g ra in b o u n d arie s . The

evidence o f low s tre n g th s o f s o f t c r y s ta ls has le d to th e conclusion

th a t c e r ta in sources of s l i p e x i s t in th e se m a te r ia l s . These sources

may have th e a b i l i t y to re le a s e d is lo c a tio n s in to th e surround ings under

sm all a p p lie d s t r e s s e s .

There a re two p o s s i b i l i t i e s f o r th e se so u rc es . They may be d i s ­

lo c a t io n s a lre a d y in th e c r y s ta ls o r , a l t e r n a t e ly , th e y may be c r y s ta l

boundaries o r im p erfec tio n s o f o th e r k inds from which d is lo c a t io n s can

be c re a te d under sm all s t r e s s e s . C o t t r e l l has a ttem p ted to ex p la in

both work h arden ing and th e B auschinger e f f e c t by th e o r iz in g th a t th ese

m obile d is lo c a t io n s became p i le d up o r trap p ed a t some b a r r i e r , i . e . ,

a g ra in boundary.

The mechanisms o f p la s t i c flow and f r a c tu re seem to have rece iv ed

most o f th e a t te n t io n in t h i s f i e l d . An a p p re c ia b le amount o f t h i s work

has been ex p erim en ta tio n w ith s in g le c r y s t a l s . One o f th e th e o r ie s o f

p la s t i c flow given by C o t t r e l l i s t h a t th e ap p aren t y ie ld s t r e s s shown

by a t e n s i l e specimen i s low er than th e a c tu a l y ie ld s t r e s s o f th e m ate r­

i a l . The y ie ld in g seems to be i n i t i a t e d a t seme lo c a t io n o f h igh s t r e s s

c o n c e n tra tio n and th en sp reads through th e m a te r ia l by th e a c t io n o f h igh

7

s t r e s s around th e y ie ld e d p o r t io n . For a s in g le iro n c r y s ta l , th e r a t i o

o f th e th e o r e t ic a l y ie ld s t r e s s to th e ap p aren t y ie ld s t r e s s v a r ie s from

two to t h r e e . Evidence th a t y ie ld in g spreads in t h i s way i s g iven by

observed su rfa ce m arkings, i . e . , Luders bands. These bands a re v is ib le

l in e s ap p earin g on th e su rfa ce o f a y ie ld e d member. The p r e f e r e n t ia l

growth o f th e se bands a long l in e s 1+5° to th e p r in c ip a l s t r e s s e s has been

used as evidence th a t o v e rs tra in e d m a te r ia l does n o t support sh ea r s t r e s s e s .

Work w ith c i r c u la r o r re c ta n g u la r b a rs in te n s io n has been con­

cerned w ith th e o ry o f f r a c tu r e , neck ing , change in u ltim a te s t r e s s and

e lo n g a tio n c h a r a c t e r i s t i c s . Jfeny a tte m p ts , bo th th e o r e t ic a l and em pir­

i c a l , have been made to p re d ic t th e magnitude o f r e s id u a l s t r e s s e s . G.

H. Handleman and W. H. Warner (7 ) have worked out d i f f e r e n t i a l equations

fo r p la s t i c work, s t r a i n , e t c . By assum ing a ra d iu s o f c u rv a tu re , P. W.

Bridgeman (8) has dev ised a form ula f o r f in d in g th e average s t r e s s

a c ro ss th e ro o t o f th e necked-down p o r tio n o f a t e n s i l e specim en. The

f a m il ia r cup and cone f r a c tu re o f a t e n s i l e specimen has been a t t r ib u te d

to a s t r e s s d i s t r ib u t io n a c ro ss th e neck . A gain, a sh ea r f a i lu r e in the

o v e rs tra in e d o u te r f ib e r s i s in d ic a te d .

Smith and Wood (3) re p o rte d X -ray d i f f r a c t io n measurements which

support th e expected s t r e s s d i s t r ib u t io n in p l a s t i c a l l y s t r a in e d round

te n s i l e specim ens. However, th e r e s id u a l s t r e s s e s th e y found a re n o t

b e lie v e d to be th e o v e ra l l la y e red s t r e s s e s caused by y ie ld in g in th e

o u te r f i b e r s . F . R. Shanley (9) re p o r ts an a p p re c ia b le d ecrease in u n i­

form e lo n g a tio n r e s u l t in g from th e s l i p r e ta rd in g p ro cesses when a ttem p ts

a re made to in c re a se th e u ltim a te s t r e n g th .

The work o f Bridgeman a ls o in c lu d ed t e n s i l e t e s t s under h igh

h y d ro s ta t ic p re s su re , th e purpose o f which was to a llo w r a d ia l s t r e s s to

8

be a c t iv e on th e o u ts id e o f th e specimen as w e ll a s w ith in . Thus f a r , no

r e la t io n s h ip has been found to p re d ic t th e a x ia l s t r e s s a c ro ss th e

c ro ss s e c tio n o f th e p l a s t i c a l l y deformed specim en.

S ev e ra l w orkers, e s p e c ia l ly those who used a com bination o f t o r ­

s io n and te n s io n , re p o r t agreem ent w ith o r a t l e a s t support o f th e Von

Mises y ie ld s tre n g th th e o ry . John Nunes (10) and some o f th e o th e rs use

B ridgem an's form ulas to determ ine a t ru e s t r e s s - s t r a i n curve f o r th e

necked down p o r tio n o f a t e n s i l e specimen and claim accu racy w ith in a

few p e rc e n t.

P rager ( l l ) has dev ised id e a l iz e d m echanical modesl to d e sc rib e

th e lo a d - s t r a in c h a r a c te r i s t i c s o f such m a te r ia ls as p e r f e c t ly p l a s t i c ,

e l a s t i c , work h a rd en in g , e t c . S t r e s s - s t r a in curves based on Bridgem an's

form ulas a re a ls o g iv en . One R ussian a u th o r , A. M. V asilev (1 2 ), c laim s >

t h a t one o f h is papers p re d ic ts th e B auschinger e f f e c t from X -ray d i f ­

f r a c t io n d a ta . No o th e r re fe re n c e s o f t h i s type were found.

CHAPTER I I I

METHOD OF APPROACH

Equipment Used

Hie equipment used fo r th e s tu d y c o n s is te d o f s t e e l specim ens,

t e s t in g machines capable o f e x e r tin g t e n s i l e and com pressive fo r c e s ,

e l e c t r i c s t r a in gages and th e a s s o c ia te d in s tru m e n ts , a b a l l j o i n t

ap p ara tu s to p rov ide a p u re ly a x ia l fo rce fo r com pression, a cu rv a tu re

m easuring a p p a ra tu s , and a su rfa ce g r in d e r .

The m ajor requ irem en ts f o r th e desig n o f th e specimens a re l i s t e d

as fo llo w s:

1 . The le n g th and th e w idth o f beams should be no g re a te r than

n e ce ssa ry in o rd e r to p rev en t ex cessiv e g rin d in g tim e .

2 . The geom etry o f the beams must perm it a s u f f i c i e n t ly t i g h t

g r ip by th e m agnetic chuck.

3 . The s iz e and type o f th e specimens must be com patib le w ith

th e t e s t in g machines a v a i la b le .

U. Hie c ro ss s e c tio n o f th e specimens should p rovide a smooth

s t r e s s flow .

5 . Hie f in is h e d specimens must be f r e e from re s id u a l s t r e s s .

The s tan d a rd round t e n s i l e specimen w ith th read ed ends was used

fo r th e s t a r t i n g p o in t o f th e d e s ig n . The beam was c u t no lo n g e r than

th e narrow se c tio n o f th e specim en. Hie beam could have been lo n g e r as

i t was d u rin g th e p re lim in a ry ru n s , b u t i t was f e l t t h a t th e m a te r ia l

in the th read ed ends, being devoid o f s t r e s s , would c o n tr ib u te n o th in g

9

10

to the measured d a ta . Hie om ission o f t h i s unneeded le n g th r e s u l te d in

a s u b s ta n t ia l re d u c tio n in m achining and g rin d in g tim e .

F igu re 3 shows th e o r ig in a l beam which was c u t from a s tan d a rd

round specim en. The m a te r ia l from th e th read ed ends was used in th i s

c a se . I t was found th a t th e rounded edge could cause some d i f f i c u l t y

when th e beam i s h e ld in th e m agnetic chuck. A lso , i t was found to be

q u ite in co n v en ien t to c u t f l a t s on th e ends o f th e beam e x a c tly even

w ith th e to p o f th e rounded edge .

S ec tio n A-A

F ig . 3 - —A Beam Cut from a S tandard Round Specimen.

I t may a ls o be seen th a t th e rounded edge would in tro d u ce a con­

s id e ra b le amount o f e r r o r i f Cut No. 2 were u sed . As a remedy f o r both

o f th e se problem s, i t was decided to use a f la t te n e d specimen which

would perm it th e c u t t in g ou t o f a beam o f re c ta n g u la r c ro ss s e c t io n .

A beam w idth o f 0 .1 in . was a r b i t r a r i l y chosen to s a t i s f y requ irem ent

No. 1 . The f l a t s on th e specimen were made 0 .3 in . wide to p rov ide a

b u f fe r a re a f o r s t r e s s s t a b i l i z a t i o n on bo th s id e s o f th e beam. The

f i n a l c o n fig u ra tio n o f th e specimens i s shown in F ig u re h .

FIG. 4 . FINAL CONFIGURATION OF SPECIMEN.

12

The t e s t in g machine used f o r th e t e n s i l e phase o f th e s tu d y was

the R ieh le 120,000 l b . T es tin g Machine which i s lo c a te d in th e C iv il

E ng ineering L abora to ry a t Brigham Young U n iv e rs ity - The m ajor advan­

tag es o f t h i s machine were smooth c o n tro l o f th e load and a s e lf -c o n ta in e d

lo a d - s t r a in p lo t t in g system .

For th e com pression phase o f the s tu d y , th e machine used was a

Baldwin-Tate-Em ery T estin g Machine, Type PTE 93, w ith 30,000 lb . capa­

c i t y . This machine i s a ls o lo c a te d in th e C iv i l E ng ineering L abora to ry

a t Brigham Young U n iv e rs ity . The s t r a in measurements in t h i s case were

made w ith e l e c t r i c a l s t r a in gages.

Three ty pes o f s t r a in gages have been t r i e d ; th e y were sim ple

paper backed gages, d u a l-w ire paper backed gages (both o f th e SR-U

v a r ie ty ) and epoxy backed g a g es . A ll th re e o f th e se ty p es may be tem­

p e ra tu re compensated fo r th e common m e ta ls . The sim ple paper-backed

gages were p re fe r re d because o f th e ease o f t h e i r i n s t a l l a t i o n and t h e i r

low p r ic e .

The cements used fo r th e se gages were Duco fo r th e paper gages

and Budd B-12 Epoxy Cement o r Eastman 910 Cement f o r th e epoxy g a g es .

The l e a s t success was experienced w ith th e Eastman 910 Cement; however,

t h i s could be due to in s u f f ic ie n t cu rin g tim e a n d /o r th e o ld age o f th e

cement on hand. The ex trem ely f a s t harden ing tim e o f th e Eastman 910

Cement i s a d e f in i t e advantage and th e p o s s i b i l i t i e s should be i n v e s t i ­

g a ted f u r th e r .

The o u tp u t o f th e s t r a in gages used was measured by an i n s t r u ­

ment which c o n s is te d o f a com bination o f a b r id g e , a t r a n s i s t o r am p li­

f i e r , and a m e te r. This in s tru m e n t, m anufactured by E l l i s A sso c ia te s ,

L td . , was d e sig n a ted BAM-1.

13

A p iece o f a p p a ra tu s u t i l i z i n g two b a l l j o in t s was d ev ised fo r

th e purpose o f p ro v id in g a n e a r ly a x ia l fo rc e f o r th e com pression phase.

A drawing showing th e c o n s tru c tio n o f i t i s g iven in F igure 5° A s a f e ty

cage made o f ^ i n . th ic k p ipe surrounds th e b a l l j o i n t a p p a ra tu s . No

i n s t a b i l i t y has y e t been encountered even though setae o f th e com pression

specimens have buck led .

A photograph o f th e cu rv a tu re m easuring ap p ara tu s i s shown in

F igu re 6 . I t c o n s is te d o f two k n ife edges to support th e beam and a

p o in ted m icrom eter to measure th e d e f le c t io n o f th e beam a t th e m idpo in t.

An ohmmeter was connected between th e m icrom eter and th e k n ife edges to

in d ic a te c o n tac t o f th e m icrom eter w ith th e beam.

'Hie g r in d e r used was a DoAll Model D-10 S urface G rin d e r. This

machine had a pow erful e lec tro -m ag n e tic chuck which was found to be

n ecessa ry to c o n tro l th e warping o f th e beams du rin g th e l a s t 0 .1 in . o f

m a te r ia l rem oval. A v is e chuck was com pletely inadequate f o r t h i s p u r­

pose . This la rg e g r in d e r a ls o had th e marked advantage o f f a s t ta b le

speeds (approx im ate ly 20 i n . / s e c . m ax.) and a 12 in . d iam eter g rin d in g

w heel which re q u ire d d re s s in g le s s f r e q u e n tly th an would be th e case w ith

a sm a lle r d iam eter w heel.

BALL JOINT SPECIMEN HOLDER FOR COMPRESSION

FIG. 5. DRAWING OF COMPRESSION APPARATUS.

F ig . 6 . —C urvature M easuring A pparatusHvn

Procedure

1.6

I t was b e lie v e d th a t any s t e e l would g ive th e d e s ire d r e s id u a l

s t r e s s e s a f t e r f i r s t b e in g p l a s t i c a l l y y ie ld e d . However, s in ce th e

Bauschinger e f f e c t was o f prime im portance, a t l e a s t th re e d i f f e r e n t

ty p es o f s t e e l were t r i e d in a search fo r t h i s phenomenon. A v e ry low-

carbon m ild s t e e l , a s t e e l w ith a mid range carbon c o n te n t, and an

e a s i ly o b ta in ed a l lo y s t e e l were in v e s t ig a te d . These th re e s t e e l s were

d e sig n a ted 1018, 1CA-5 and l l¥ j- . On the b a s is o f showing a r e s id u a l

s t r e s s p a t te rn and e x h ib it in g the B auschinger e f f e c t , each o f th e se th re e

s te e l s seemed to be e q u a lly s u i te d . The 10l8 s t e e l was a r b i t r a r i l y

chosen. A p a r t i a l s t r e s s s t r a i n diagram fo r annealed 1018 s t e e l i s

g iven in F igure 7.

Both e l e c t r i c s t r a in gages and th e d i f f e r e n t i a l tran sfo rm er ex­

ten s io m ete r were used; th e re fo re , i t was n e ce ssa ry to compare th e o u tpu t

o f each w ith a known s ta n d a rd . Hie d i f f e r e n t ia l - t r a n s f o r m e r s t r a in

m easuring system was c a l ib ra te d w ith an e l e c t r i c r e s is ta n c e w ire s t r a in

gage which in tu rn was c a l ib ra te d w ith a m echanical d ia l-g a g e e x te n s io ­

m e te r. A more d i r e c t method was p re fe r re d , b u t m echanical d i f f i c u l t y

was encoun tered . When h igh ranges o f s t r a i n were measured w ith the

e l e c t r i c s t r a in gages, th e re was some d i f f i c u l t y encountered in o b ta in ­

in g a s u f f i c i e n t ly h igh c a l ib r a t io n p o in t on th e m eter s c a le . The ga in

c o n tro l on th e b rid g e would n o t go low enough to perm it th e d e s ire d

c a l ib r a t io n p o in t . This was e a s i ly rem edied by adding a 10,000 ohm

p o ten tio m e te r in s e r ie s w ith th e common w ire which le d to both th e

a c t iv e and th e com pensating g a g es . No adverse e f f e c t on th e m easure­

ments r e s u l te d .

The la b o ra to ry techn iques used fo r th e i n s t a l l a t i o n o f th e s t r a in

gages have been in c lu d ed fo r convenience.

IT

The c o n s is te n t ly su c c e ss fu l method f o r i n s t a l l i n g th e s t r a in

gages i s o u tlin e d as fo llo w s: F i r s t th e specimen i s c a r e f u l ly p o lish ed

to remove a l l oxide and to o l m arks. I t i s th en c leaned by swabbing

w ith a m ixture o f 1 p a r t (by volume) xylene and 2 p a r ts m e th y l-e th y l-

key tone. This c lean in g s te p could j u s t as w e ll have been done by

swabbing f i r s t w ith to lu o l and then ace to n e . Roughening th e su rface

o f the specimen w ith 180 to 220 g r i t emery c lo th i s a ls o recommended.

When paper gages a re i n s t a l l e d , a l i b e r a l co a t o f Duco cement

i s a p p lie d to each s id e o f th e gage. The a i r bubbles and excess cement

a re c a r e f u l ly worked away from th e gage by firm f in g e r p re s su re . A

p iece o f foam ru b b e r, p ro te c te d by a s t r i p o f waxed pap er, i s used to

a p p ly p re ssu re to th e gage du rin g th e i n i t i a l s tag e o f th e d ry in g . Hie

p iece o f foam rubber should overlap th e gage by a t l e a s t l / 8 in . on a l l

s id e s . Tape, a w eight o r a clamp may be used to ho ld th e sponge rubber

in p la c e . The p re ssu re should be 1-2 p s i . S ince Duco i s a so lv e n t-

escape cement, no covering should be l e f t over th e gage fo r more than

20 m in u te s .

When epoxy gages a re i n s t a l l e d , th e cem enting procedure i s

s l i g h t l y more in v o lv ed . A fte r th e su rfa ce o f th e specimen i s roughened

and, o f co u rse , c leaned a g a in , a l i g h t co a t o f Budd M etal C ond itioner

No. 1 i s a p p lie d to th e specim en. This s o lu tio n i s a p p a re n tly a m ild

o x id iz in g ag en t s im ila r to gun b lu e in g . The s te p j u s t m entioned a p p lie s

only to fe rro u s m e ta ls .

The back s id e o f th e epoxy gages should be c a r e f u l ly roughened

w ith an ex trem ely f in e emery pap er, i . e . , 500 g r i t p ap er. Care must be

taken to avo id sanding through to th e m eta l f i lm . When Budd B-12 Cement

i s u sed , th e foam rubber p re ssu re pads should be l e f t in p lace du rin g

18

the e n t i r e 12 hour cu rin g p e r io d . When Eastman 910 Cement i s u sed , an

a c c e le r a to r may he coated on th e gage, and th e cement w i l l then cure

a lm ost im m ediately.

During th e e a r ly p a r t o f t h i s p e rio d o f re s e a rc h , e l e c t r i c i a n ’s

p la s t i c tap e was t i g h t l y wrapped around th e gage a f t e r th e cement had

cu red . This was an a ttem p t to p ro te c t th e gage bo th from m oistu re and

m echanical damage. A comparison o f th e accu racy o f tap ed and untaped

gages showed th i s p ra c t ic e o f ta p in g to he q u ite d e tr im e n ta l .

One o f the g r e a te s t su sp ec ts o f experim en tal e r r o r in work of

t h i s k ind i s bending caused by the load -p roducing a p p a ra tu s . This bend­

in g would n o t only a f f e c t th e r e s id u a l s t r e s s m easurem ents, b u t would

a ls o in f lu e n c e th e column a c tio n o f th e specimen d u rin g com pression, thus

causing prem ature b u ck lin g . Means were d ev ised to reduce th e e f f e c t s

o f t h i s bending from app rox im ate ly 15 to 3 per c e n t.

The s p h e r ic a l jo in t s in th e t e n s i l e f ix tu r e s o f th e t e s t in g

machine were f i r s t c lean ed , lapped and lu b r ic a te d . This d id n o t reduce

th e bending in te n s io n to an accep tab le v a lu e , so th e a x is o f average

d e f le c t io n fo r te n s io n was found. With th e lo c a t io n o f t h i s a x is known,

the beams to be used f o r d is s e c t io n were c u t from t h i s most fav o rab le

p o s i t io n . C hapter IV g ives an ex p lan a tio n o f t h i s c a lc u la t io n . The

term , a x is o f average d e f le c t io n , i s used because th e e x p re ss io n , n e u tr a l

a x is , does n o t r i g i d ly ap p ly to a s t r e s s f i e l d composed o f te n s io n and

ben d in g .

The bending du rin g th e se t e s t s was e v a lu a ted by making s t r a in

measurements a t U lo c a t io n s sym m etrically chosen around th e p e rip h e ry

o f a c i r c u la r t e s t specim en. A photograph o f a ty p ic a l round specimen

w ith fo u r s t r a in gages a tta c h e d i s shown in F igu re 8 . For convenience.

19

th e fo u r compass d ir e c t io n s were chosen as th e lo c a t io n s fo r s t r a in

measurement. In accu racy caused by th e in d iv id u a l e r ro r s o f th e gages

and e c c e n t r ic i ty o f th e specimen was minimized by r o ta t in g th e specimen

in i t s h o ld e rs 90° a f t e r each run u n t i l each compass d i r e c t io n had been

occupied by each o f th e fo u r gages.

A sm all amount o f bending (approx im ate ly 8$) was found to e x is t

in th e specimen du rin g com pression t e s t s , so ag a in th e a x is o f average

d e f le c t io n was c a lc u la te d . Ihe com plete system used f o r com pression i s

shown in F igu re 9-

20

FIG. 7. PARTIAL STRESS STRAIN DIAGRAM FOR ANNEALED 1016 STEEL.

NOTE * THE TEST WAS TERMINATED BEFORE THE NECKING POINT WAS REACHED.

21

F ig . 8 . —Round Specimen w ith Four S tr a in Gages A ttach ed .

ioF ig . 9»—System Used f o r Compression

Freedom from re s id u a l s t r e s s in th e specimens was a ssu re d by-

hav ing them annea led in an i n e r t atm osphere a t a commercial h e a t t r e a t ­

in g f a c i l i t y .

I t was co n sidered d e s ir a b le t o s t r a i n th e specimens to ap p ro x i­

m ate ly f iv e d i f f e r e n t p o in ts on th e s t r e s s - s t r a i n d iagram . The p o in ts

were a r b i t r a r i l y chosen. The p o in t used f o r th e c o n tro ls was a p p ro x i­

m a te ly halfw ay up th e e l a s t i c p o r tio n o f th e t e n s i l e s id e o f th e d ia ­

gram. Two more p o in ts were chosen in th e work h arden ing reg io n o f th e

t e n s i l e s id e . The p o in t o f u ltim a te s tre n g th was n o t used because i t

was b e lie v e d th a t th e most pronounced extrem es in th e r e s id u a l s t r e s s

p a t te rn s would occur a t seme p o in t e a r ly in th e work harden ing reg io n

where a l l f ib e r s had n o t y ie ld e d th e same amount.

A ll o f th e com pression specimens were f i r s t p u lle d to p o in t No.

3 , th e maximum te n s i l e s t r a in used , and th en compressed to one o f th e

two com pression p o in ts u sed . Most o f th e specimens cou ld be compressed

beyond th e 5 ,000 p i n / i n p o in t even though a few o f them d id b u c k le .

I f more specimens had been a v a i la b le , a n o th e r p o in t a t a h ig h e r com­

p re s s iv e s t r a i n would have been u sed . A lso , a p o in t o f pure compres­

s iv e s t r a in (w ithou t p r io r t e n s i l e s t r a i n ) would have been used f o r

purposes o f com parison.

The p o in ts o f s t r a i n which were used a re l i s t e d a s fo llo w s:

P o in t No. S tr a in Used p, i n . / i n .

123k5

Tension, TOO C ontro ls T ension, 19,300 T ension , 25,200T ension, 25,200; Compression, 3 ,000 T ension, 25,200; Compression, 5 ,000

This in fo rm atio n i s p re sen ted g ra p h ic a l ly in F igure 10.

502k

FIG. 10. DIAGRAM OF STRAIN VALUES USED.

25

A fte r th e specimens were s t r a in e d in th e t e s t in g m achines, th e y

were mounted in th e m ill in g m achine. S len d er re c ta n g u la r beams were

th en c a r e f u l ly c u t ou t o f th e specimens by u s in g a m il l in g c u t te r and

a p a r t in g saw. The lo c a t io n o f th e se beams i s shown in F igu re 11.

Beam

The dep th o f th e m il l in g c u ts was l im ite d to 0.015 in . to minimize

m achining s t r e s s e s . The s id e s o f th e beams were then l i g h t l y ground

to p rov ide a uniform th ic k n e ss and to ag a in reduce th e e f f e c t s o f the

m il l in g s t r e s s e s . The dep ths o f th e g rin d in g c u ts d id n o t exceed 0.0005

in .

The beams c a r r ie d th e same id e n t i f i c a t io n numbers a s th e p a re n t

specim ens. These numbers were l i g h t l y sc r ib e d on th e s id e s o f th e beams

n e a r th e ends. The ends a ls o c a r r ie d group* m arkings, i . e . , end b e v e l,

c h is e l p o in t , d iag o n a l s c ra tc h e s , e t c . With t h i s system o f p o s i t iv e

i d e n t i f i c a t io n , i t was p o s s ib le to g rin d a l l o f th e beams a t th e same

tim e . The d a ta from each beam was reco rded s e p a ra te ly . F in a l ly , when

the g rin d in g and d e f le c t io n m easuring were f in is h e d , a common th ic k n e ss

was chosen as a s t a r t in g p o in t and th e d e f le c t io n s fo r a l l o f th e beams

in th e group were averaged a t each succeeding th ic k n e s s .

group c o n s is te d o f th ose specimens s t r a in e d to th e same p o in t on th e s t r e s s - s t r a i n cu rv e .

26

The procedure o u tlin e d by th e G eneral Motors Research L ab o ra to r­

ie s (2) was s t r i c t l y fo llow ed w hile th e beams were be ing ground. Any

g rin d in g cu ts eq u al to o r g re a te r than 0.0005 i n . may be expected to in ­

tro d u ce r e s id u a l s t r e s s e s . M. I . H etenyi (13) g ives seme in te r e s t in g

v a lu es f o r r e s id u a l s t r e s s e s caused by g r in d in g . A copy o f th e p ro ­

cedure f o r beam d is s e c t io n can be found in th e appendix o f t h i s t h e s i s .

A ll tw enty specimens were ground a t once; approx im ate ly te n

hours were re q u ire d fo r each o f th e fo u r te e n d e f le c t io n measurements ta k en .

The s ig n convention fo r th e c u rv a tu re was the same one used by

Henry S. Todd ( l 4 ) . When th e edge being ground away was p laced n ex t to

th e gage, a convex-downward cu rv a tu re was regarded a s p o s i t iv e . The i n ­

board edge o f th e beam i s u s u a lly ground away f i r s t . The g rin d in g of

t h i s edge f i r s t has been d esig n a ted "Cut No. 1 ." O ccasio n a lly , i t i s

d e s ir a b le to change to Cut No. 2 and g r in d th e outboard edge. This i s

n e c e ssa ry when th e cu rv a tu re o f th e beams causes d i f f i c u l t y in h o ld in g

th e beams on a m agnetic chuck. Only Cut No. 1 was used in t h i s s tu d y .

The d e f le c t io n s were m easured on bo th s id e s o f each beam and th e average

o f th e two measurements was used fo r cu rv a tu re and s t r e s s c a lc u la t io n .

A sample d a ta sh e e t i s g iven in F igu re 12.

The m athem atical r e la t io n s h ip s between th e removed s t r e s s e s and

th e c u rv a tu re changes in th e beams i s given in Appendix A. Hand c a lc u ­

l a t io n o f th e se s t r e s s e s i s an ex trem ely tim e-consum ing p ro c e ss . Henry

S. Todd (lU ) has p repared a program f o r th e c a lc u la t io n o f th e r e s id u a l

s t r e s s e s on a d i g i t a l cam putor. A s l i g h t l y m odified v e rs io n o f t h i s p ro ­

gram was used in t h i s s tu d y . A copy o f t h i s m odified program i s given

in Appendix E.

27

Date:

Spec. No.:

Beam S ize :

I n i t i a l T h ickness:

Cut No.:

August 19, 1961+

52

0 .2 x 0 .1 x 3-3 in .

O.213I* in .

l

Thickness Inboard Outboard Mean

in . M icrometerReading

A( la r g e r +)

M icrometerReading

A( la rg e r - )

A

0.1938 0.5000 0.0000 0 .1*996 0.0000 0.0000

O.1838 0.5002 0.0002 0 .1*998 -0 .0002 0.0000

O.1676 0.5000 0.0000 0.5000 -o.oooi* -0.0002

0.1500 0.1*998 -0.0002 0.5000 -o.oooi* - 0.0003

0.1332 0.1*996 -o.oooi* 0 .1*998 -0 .0002 - 0.0003

0.1176 0 .1*992 -0 .0008 0 .1*998 -0 .0002 - 0.0005

0.101U 0 .1*990 -0 .0010 0.5000 -o.oooi* - 0.0007

0.0852 0.14-988 -0.0012 0.5002 -0.0006 -0.0009

O.0696 0 .1*981* -0.0016 0.5001* -0.0008 -0.0012

0.0U86 0,1*970 -0 .0030 0.5016 -0 .0020 -0.0025

0.0380 0.1*956 -0.001*1* 0.5030 - 0 . 0031* - 0.0039

0.0036 0.1*91*8 -0.0052 0 . 501+0 -0.001*1* -o.ooi*8

0.0282 O.I+92I* -0.0076 0.5062 -0.0066 - 0.0071

A ll measurements a re in in c h e s .

F ig . 1 2 .—Sample D e fle c tio n Data S h ee t.

Data Reduction

28

Because o f th e n a tu re o f th e d a ta measured in t h i s work, th e re

e x is t s a need f o r one o r more s te p s o f d a ta sm oothing. As th e procedure

now s ta n d s , a d e f in i te smoothing ta k es p lace th re e tim es be fo re th e

s t r e s s e s a re c a lc u la te d .

When th e d e f le c t io n o f th e beams was m easured, read in g s were taken

on both th e to p and th e bottom o f th e beam. The average o f th e two values

i s re co rd ed . There i s a p ro v is io n in th e computer program to use d e f le c ­

t io n measurements from bo th s id e s s e p a ra te ly , b u t i t was found to be

exped ien t to average the two va lu es and in p u t th e d a ta on ly once to th e

com puter. The sample d a ta s h e e t. F ig u re 12, shows th a t a co n sid e rab le

amount o f smoothing took p lace in t h i s s te p .

When a beam r e s t s on th e c u rv a tu re m easuring gage, th e w eight o f

th e overhanging ends causes a s l ig h t e r r o r in th e c u rv a tu re . This may be

c o rre c te d by th e in p u t o f a c o n sta n t to th e s t r e s s c a lc u la t io n program.

The use o f t h i s c o n sta n t i s ex p la in ed in Appendix F . However, t h i s e r ro r

becomes s e l f - c o r r e c t in g when th e cu rv a tu re i s measured on bo th s id e s of

th e beam. A lso , th e e r r o r caused by a s l ig h t b u rr on one s id e o f the

beam i s reduced to o n e -h a lf by t h i s method.

The second smoothing o f th e d a ta was accom plished by th e av erag in g

to g e th e r o f th e d e f le c t io n s o f th e 2 to 6 specimens in each group.

O bviously, i t was d e s ire a b le to have a s many specimens in a group

a s p o s s ib le . When th e d a ta from each of th e beams in a group i s averaged

to g e th e r , i t i s hoped th a t many of th e in d iv id u a l d if f e re n c e s would d i s ­

ap p ea r. Inc luded would be sm all amounts o f e c c e n t r ic i ty , unusual machin­

ing s t r e s s e s , e t c .

29

The t h i r d smoothing o f th e d a ta ta k es p lace when th e computer

program a p p lie s a curve f i t t i n g procedure to th e in p u t d a ta . The r e l a ­

t i v e l y com plicated s t r e s s com putation , th e n , ta k es p lace on in p u t d a ta

which i s known to be smooth. This saves machine tim e and improves the

accu racy o f th e o u tp u t. A lso , w ith t h i s method, i t i s much e a s ie r fo r

th e program u se r to check th e accu racy o f th e curve f i t . The cu rv a­

tu re : o f th e beam as measured may be q u ic k ly c a lc u la te d from th e r e l a ­

t io n s h ip :

C 2A

1 + A(1)

A i s th e d e f le c t io n o f th e c e n te r o f th e beam measured in in c h es ,

and C i s th e cu rv a tu re in in c h es”1 .

o r C « 2A fo r A ^ 0.0075 in . (2)

When th e computed s t r e s s e s were f i r s t re c e iv e d from th e computa­

t io n s e c t io n , th e q u es tio n im m ediately a ro se concern ing which o f th e

eq uation o rd ers gave the b e s t curve f i t . The obvious s o lu t io n would be

to p lo t th e cu rv a tu re c a lc u la te d from o r ig in a l d a ta on th e same p iece o f

paper w ith some o f th e curves produced by th e curve f i t r o u t in e . The

curve w ith th e c lo s e s t resem blence to th e o r ig in a l d a ta would be chosen.

The program a ls o p r in t s th e s tan d a rd d e v ia t io n o f th e d a ta , which i s an

index o f th e curve f i t e r r o r .

I t was l a t e r n o tic e d th a t th e p lo t o f beam cu rv a tu re v s . d is ta n c e

produced a curve which was n o t r e a d i ly approxim ated by th e l e a s t squares

curve f i t p ro ced u re . That i s to say th a t th e curve was n o t e a s i ly re p re -

2 3sen ted by th e polynom ial C = A + Bx + Cx + Dx + . . . Because o f t h i s ,

th e s tan d a rd d e v ia t io n , cr, d id n o t n e c e s s a r i ly in d ic a te th e q u a l i ty o f

f i t between th e measured and th e computed c u rv a tu re . O bviously , th e

30

h ig h e r o rd ers o f f i t t i n g cu rv a tu re s came c lo s e r to th e measured p o in ts ,

b u t th e d ip s in th e f i t t e d curves seemed to in tro d u ce undue e r r o r . At

th e p o in t o f g r e a te s t th ic k n e s s , th e f i t t e d curve in v a r ia b ly assumed a

s lope q u ite d i f f e r e n t from th e n e a r ly zero s lope o f th e measured d a ta .

The r e s u l t was th e f a ls e in d ic a t io n o f h ig h - s t r e s s le v e ls in the s p e c i­

men. An example o f some measured c u rv a tu re s and th e f i t t e d curve i s

shown in F igu re 13. The above polynom ial was th e one employed by th e

curve f i t t i n g ro u tin e o f th e s t r e s s c a lc u la t io n program used . I f more

tim e had been a v a i la b le , th e program would have been re w r it te n w ith a

d i f f e r e n t techn ique fo r curve f i t t i n g .

CURV

ATUR

E ,

IN.

31

— SECOND ORDER— FOURTH ORDER

FIG. 13. MEASURED AND F IT T E D CURVATURE FOR TH E P O IN T NO. 3 B E A M S.

32

S ev era l rem edies fo r t h i s s i tu a t io n were co n sid e red . The most

prom ising seemed to he a method where second o r r e r eq u a tio n s were f i t t e d

to sm all groups o f th e measured c u rv a tu re s . The tech n iq u e used fo r the

c o n tro ls was as fo llow s:

1 . The f i r s t f iv e p o in ts were f i t t e d w ith a q u a d ra tic eq u a tio n ,

2 . The f i r s t th re e p o in ts were skipped and th e nex t f iv e p o in ts

were f i t t e d .

3- This process was con tinued u n t i l th e e n t i r e curve had been

com pleted.

U. The cu rv a tu re v a lues a t th e overlapp ing p o in ts were then

av erag ed .

5 . The r e s u l t in g com posite curve was then used as in p u t fo r the

s t r e s s c a lc u la t in g ro u t in e .

When th e r e s u l t s o f t h i s method were compared to those o f the

method used in F igure 13, i t was found th a t th e e r ro rs due to curve f i t t i n g

had decreased to approx im ate ly 5^ o f t h e i r form er v a lu e s .

V a ria tio n s o f th e method ju s t d esc rib ed were t r i e d fo r each of

the f iv e groups o f beams. S tre s s curves were p lo t te d in each in s tan ce

to re v e a l th e t r e n d s . S ince th e d a ta from th e v a rio u s groups was found

to respond to d i f f e r e n t te ch n iq u es , t h i s method was continued u n til , s u f ­

f i c i e n t l y smooth s t r e s s curves were o b ta in ed . F igure lU shows a com pari­

son o f th e two systems used to o b ta in s t r e s s d a ta fo r th e c o n t r o ls .

33

DISTANCE IN IN.

----- CONVENTIONAL CURVE FIT- - - - - - C URVE F I T T E D WITH G R O U P S O F P O I N T S (TREND). . . . . . . . 4 P T S . U S E D AT A TIM E j I PT. O VERLAPPED

. = 5 USED } 2 OVERLAPPED. AND 4 U SE D } 2 O V E R L A P P E DFIG. 14 . R E S ID U A L ' S T R E S S . V S . DISTANCE FOR THE C O N T R O L S.

CHAFFER IV

DISCUSSION OF THE MEASUREMENTS AND RESULTS

E ffe c t o f E c c e n tr ic i ty

The magnitude o f th e bending caused by th e load -p roducing ap p ara ­

tu s may be b e t t e r a p p re c ia te d by co n sid e rin g th e fo llo w in g c a lc u la t io n

o f s t r e s s as a fu n c tio n o f e c c e n t r ic i ty and lo a d .

L et: S

A

P

M

c

I

e

th e maximum t o t a l s t r e s s

c ro ss s e c t io n a l a re a o f th e specimen

a p p lie d load

moment caused by e c c e n t r ic i ty

maximum d is ta n c e from th e n e u tr a l a x is

moment o f i n e r t i a

e c c e n t r ic i ty

P . Me to 6h Pe . 5 .S = A + T “ T 2 " + — TT 2 ( 3 )

( ! + - - ) * K(1 + l 6 e)

bending s t r e s s _ l 6e t o t a l s t r e s s ~ 1 + l 6e (h)

A graph o f t h i s ex p ress io n i s shown in F ig u re 15.

3^

35

F ig . 1 5 .—Bending v s . E c c e n tr ic i ty .

C a lcu la tio n o f th e Axis o f Average D e fle c tio n

I f a p la n a r c ro ss s e c tio n o f th e specimen i s assumed to rem ain a

p lane du rin g e l a s t i c defo rm ation , th e a x is o f average d e f le c t io n o f th e

te n d in g f i e l d may he e a s i ly found w ith a n a ly t ic geom etry. The s te p s a re

o u tlin e d as fo llo w s:

1 . Let th e X-Y p lane he th e u n s tra in e d c ro ss s e c tio n o f th e

specim en.

2 . Let th e Z co o rd in a te he in th e d i r e c t io n o f th e measured

s t r a in a t a g iven p o in t .

3 . Solve fo r th e l in e o f in te r s e c t io n between th e in c l in e d plane

o f s t r a in and th e X-Y p la n e . This l in e i s th e a x is o f

average d e f le c t io n .

36

U. O btain th e eq u a tio n and hence th e d i r e c t io n o f th e a x is o f

average d e f le c t io n .

As an example, p a r t o f th e a v e ra g e -d e f le c t io n a x is c a lc u la t io n

fo r th e R ieh le T e s te r w i l l he g iv en .

Using th e sc a le :

1/ V on th e specimen equals u n ity

10 p, in / i n o f s t r a in equals u n ity

and av erag in g th e b values: o f s t r a i n f o r each compass d i r e c t io n ,

fo llo w in g co o rd in a tes a re o b ta in e d .

D ire c tio n X Y z *

N 0 1 -3 .6 3

E 1 0 - 0.63

S 0 -1 2 .87

W -1 0 1 .37

I f th e u su a l eq uation f o r a p lane i s used , th e fo llow ing system

i s produced:

D = 0 (5)Ax + By + Cz

1 . OA +B -3.630

0119

2 . A +0B -0.63C

0II9

3 . OA -B +2.87C +D = 0

b . -A +0B +1.37C

0119

I f th e se equations a re so lved sim u ltan eo u sly in groups o f 3 > and

i f th e r e s u l t s a re averaged .

vThe v a lu es fo r Z re p re se n t d if f e re n c e s from th e mean.v /

Since

37

A = 1 .00 C

B = 3 .25 C

D = 0 .0

Ax + By + Cz + D = 0 fo r one plane

and

Y = ± B

Or, in t h i s c a se .

Cz

• X.

= 0 fo r th e o th e r . (6 )

(7)

y = - 0 . 308X.

The a x is o f average d e f le c t io n w i l l he as shown in F igure l6 .

N

F ig . l 6 . —The Axis o f Average D e fle c tio n in Tension

fo r th e R ieh le T e s te r

In a s im ila r manner th e lo c i o f th e axes o f average d e f le c t io n

were c a lc u la te d fo r th e Baldwin T e s te r in te n s io n and th e h a l l j o i n t com­

p re ss io n ap p ara tu s when mounted in th e Baldwin T e s te r . They a re shown in

F igure 17.

This p a r t i c u la r machine had one v a rn ish ed jaw which was lo c a te d in the low er p o s i t io n .

38

Com pression^ Tension

F ig . 1 7 .—Axes o f Average D e fle c tio n fo r th e

Small Baldwin T es tin g Machine.

,f >l<:The sc r ib e d l in e on th e com pression ap p ara tu s was on th e E ast s id e .

The w e ll- lu b r ic a te d b e a rin g b a l l s in th e com pression ap p ara tu s

should have p reven ted th e tra n sm iss io n o f any bending moment in to th e

com pression specim en. This would have been th e case i f th e b a l l s u r ­

faces were f r i c t i o n l e s s and i f th e c e n te rs o f th e b a l l s la y e x a c tly on

th e a x is o f th e specim en. A pparen tly , some e c c e n t r ic i ty d id e x i s t , and

a sm all amount o f bending was p re s e n t . For t h i s re a so n , th e a x is o f

average d e f le c t io n was c a lc u la te d f o r the b a l l j o i n t a p p a ra tu s .

I n te r p r e ta t io n o f th e Measurements

Some d i f f i c u l t y was encountered w ith th e s t r a in gages a t th e b e ­

g inn ing o f t h i s p ro je c t ; in c o n s is te n t r e s u l t s from one gage in s t a l l a t i o n

to th e n ex t f r u s t r a te d a l l a ttem p ts to make p re c ise m easurem ents. The

beh av io r o f u n fa m ilia r s t r e s s s t r a i n curves made p re c is e s t r a in in g o f

th e s t e e l specimens q u ite d i f f i c u l t .

39

A d e f in i te n o n - l in e a r i ty i s u s u a lly observed in th e f i r s t 150

m icro -inches p e r inch o f s t r a i n . A pparen tly , t h i s in accu racy i s caused

by th e f i n i t e th ic k n e ss o f th e cement la y e r under th e gage. Whenever a

h igh degree o f accu racy i s d e s ire d , an i n i t i a l read in g a t the 150 (j,

i n . / i n . le v e l may be made and t h i s amount i s th en s u b tra c te d from a l l

subsequent re a d in g s . C o rrec tio n i s u s u a lly made by s h i f t in g th e s t r e s s

s t r a in curve to cause i t to pass through th e o r ig in .

When th e y ie ld p o in ts* were being c a r e f u l ly m easured, some r a th e r

unusual beh av io r o f th e s t r e s s s t r a i n diagram was n o tic e d . When th e

t e s t in g machine was stopped a t th e t e n s i l e y ie ld p o in t , th e specimen

im m ediately unloaded to a p o in t approx im ate ly -j- o f th e way down th e

l in e a r p o rtio n o f th e s t r e s s - s t r a i n diagram . When th e machine was ag a in

tu rn ed on, th e specimen suddenly began to c reep u n c o n tro lla b ly a t a

n e a r ly co n s ta n t lo a d . T his y ie ld in g was q u ite re m in isc en t o f th e b e ­

h a v io r o f h o t g la ss or some o th e r supercoo led l iq u id . A rough sketch

o f t h i s b ehav io r i s shown in F igure 1 8 .

F ig . 1 8 .—The s t r e s s - s t r a i n diagram in th e

v i c in i t y o f th e y ie ld p o in t .

*The term " y ie ld p o in t" used in t h i s s tu d y was a r b i t r a r i l y d e fin ed as th e p ro p o r tio n a l l i m i t .

^0

The most lo g ic a l ex p lan a tio n fo r th e f i r s t phenomenon stems from

a c o n s id e ra tio n o f th e t e s t in g machine i t s e l f . As would he expected ,

th e machine i s h u i l t q u ite rug g ed ly , and th e sp rin g co n s ta n ts th ro u g h ­

out i t s s t ru c tu re a re h ig h . When th e specimen y ie ld s s l i g h t ly , th e

change in d e f le c t io n would cause an a p p re c ia b le change in load due to

th e h igh o v e ra l l sp rin g c o n s ta n t o f th e t e s t e r i t s e l f . A sp rin g c o n sta n t7

capable o f causing such an e f f e c t would have to be in th e o rd e r o f 10

l b s . / i n . This f ig u re i s q u ite r e a l i s t i c f o r a t e s t in g machine o f th i s

s i z e . This e f f e c t o f un load ing down th e e l a s t i c curve p robab ly has n o t

been n o tic e d b e fo re because du ring t e n s i l e t e s t s , i t i s u s u a lly unnec­

e s s a ry to s to p th e machine w hile read in g s a re ta k en .

A c a r e fu l check o f o th e r s t r e s s - s t r a i n diagram s has shown th a t

th e o th e r phenomenon, th e u n c o n tro lla b le c reep im m ediately a f t e r y ie ld ,

i s q u ite common and was s a l i e n t because o f th e h igh s e n s i t i v i t y o f the

measurements being ta k en .

In te r p r e ta t io n o f th e R esu lts

Since th e law o f s t a t i c e q u ilib riu m must be s a t i s f i e d , every

r e s id u a l s t r e s s must be ba lanced by a s t r e s s o r group o f s t r e s s e s o f n e t

op p o site s ig n . A c tu a lly , i t i s th e fo rce s which must be b a lan ced , so

th e s t r e s s e s must be in te g ra te d over t h e i r a re a s o f in flu en c e b efo re

th e y may be p ro p e rly co n sid e red . F ig u re ih shows a preponderance o f

t e n s i l e s t r e s s (e s p e c ia l ly a t th e outboard edge) which i s p h y s ic a lly

im p o ssib le . The o n ly lo g ic a l e x p lan a tio n f o r t h i s phenomenon i s th a t

l i g h t t e n s i l e s t r e s s e s were s te a d i ly in tro d u ced through th e g rin d in g

p ro c e ss . O therw ise, th e p o s i t iv e a re a under th e curve would n e a r ly

equal th e n eg a tiv e a r e a .

h i

The s t r e s s curves fo r p o in t No. 3 , F igure 19B, show te n s i l e

s t r e s s e s a t th e c e n te r o f th e specimen and com pression n ear th e o u ts id e

as would be expected i f th e o u te r f ib e r s y ie ld e d f i r s t . I f t h i s s t r e s s

p a t te rn con tin u es to e x i s t and i f th e B auschinger e f f e c t con tinues to

e x i s t f o r specimens s t r a in e d to h ig h e r le v e ls o f te n s io n , th e evidence

may be co n sidered to be v e ry s tro n g th a t th e r e s id u a l s t r e s s e s a re th e

cause o f th e B auschinger e f f e c t . No conclusion can be drawn in t h i s case

because th e s t r a in s used were r e l a t i v e ly sm all compared w ith th e s t r a in

a t th e p o in t o f u ltim a te s t r e n g th . I f f u r th e r ex p erim en ta tio n a t p o in ts

o f much h ig h e r s t r a i n shows a d isappearance o f th e s t r e s s p a t te rn in

F ig u re 19B and a c o n tin u a tio n o f th e B auschinger e f f e c t , t h i s ex p lan a tio n

w i l l have been proven in v a l id .

F ig u re 19 a ls o shows a g re a te r p o s i t iv e a re a under th e cu rv e . A

sm a lle r a re a a t th e ou tboard s id e would be expected in a c i r c u la r s p e c i­

men, b u t s in ce th e se specimens were f l a t t e n e d , equal a re a s would be ex­

p e c te d . A gain, th e g re a te r p o s i t iv e a re a seems to in d ic a te th e in tro d u c ­

t io n o f t e n s i l e s t r e s s by g rin d in g .

F igu re 20B shows d e f in i te com pressive s t r e s s e s on th e in s id e and

t e n s i l e s t r e s s e s on th e o u ts id e o f th e com pression specim ens. This in ­

d ic a te s t h a t th e outerm ost f ib e r s were th e f i r s t to y ie ld due to th e

h ig h e r com pressive s t r e s s e s . As th e load was re le a s e d , th e p a r t i a l l y

unloaded o u te r f ib e r s were fo rced in to te n s io n by th e s t r a in r e la x a tio n

in th e r e s t o f th e specim en. These t e n s i l e s t r e s s e s a re balanced by the

com pressive s t r e s s n e a r th e c e n te r .

F igu re 2QA, th e p lo t fo r th e specimens o f p o in t No. k , shows the

same type o f s t r e s s p a t te r n , b u t n o t so v iv id ly . A lso , a su g g estio n o f

th e p o in t No. 3 p a t te rn i s a p p a re n t. I t may be concluded th a t a t t h i s

k2

p o in t th e t r a n s i t i o n ta k es p lace where th e t e n s i l e p a t te rn s d isap p ea r

and g ive way to th e com pressive p a t te r n s . A gain, as in a l l o f th e s t r e s s

p a t te r n s , th e b ia s o f th e t e n s i l e g rin d in g s t r e s s e s i s p re s e n t .

F ig u res 19A through 2GB must be accep ted w ith some re s e rv a tio n

because in each case th e s c a t t e r o f th e p rim ary curves was approx im ate ly

tw ice as severe as i t was in F igu re lU , Trends were in d ic a te d b u t no t

a s d e f in i t e ly as th e y were in th e case o f F igure lU .

STRE

SS

IN

RS.I.

X |

Q__

__

STRE

SS

IN

P.S.I.

^3

x 4 USED I OVERLAPPED

B. POINT NO. 3

FIG. J 9 . RESIDUAL S T R E S S P A T T E R N FOR T E N SIL E SPE C IM E N S

STRE

SS

IN

PSI

kb

* ' A 5. USED 2 OVERLAPPEDA. POINT NO. 4 ° _ 4 U SE D 2 OVERLAPPED

4 U SE D_ I OVERLAPPED0

B. POINT NO. 5FIG. 2 0 . RESIDUA!_ _ S T R E S S PA T T E R N F O R ‘..COMPRESSION SPE C IM E N S.

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

The in fo rm atio n gained from th e specimens s t r a in e d on ly in te n s io n

was n o t c o n c lu s iv e , b u t th e p o in t No. 3 d a ta showed a s t r e s s p a t te rn which

seemed to su p p o rt th e "Bauschinger e f f e c t from re s id u a l s t r e s s " th e o ry .

These specimens which were p u lle d to th e h ig h e s t t e n s i l e s t r a i n showed

th a t y ie ld in g does occur f i r s t in th e o u te r f ib e r s and th a t th e se p a r ­

t i a l l y unloaded f ib e r s a re pushed in to com pression a s th e e x te rn a l lo ad

i s removed.

The same type o f phenomenon occurred in th e com pressive specim ens.

The h ig h e r - s tr e s s e d o u te r f ib e r s y ie ld e d f i r s t and th en were fo rced in to

te n s io n a s th e com pressive e x te rn a l lo ad was re le a s e d . A gain, th e e f f e c t

was more pronounced in th e more h ig h ly s t r a in e d specim ens. In bo th i n ­

s ta n c e s , i t does seem th a t th e B auschinger e f f e c t i s caused by re s id u a l

s t r e s s . Repeated p la s t i c s t r a in in g o f s t e e l in one d ir e c t io n would cause

r e s id u a l s t r e s s e s which would ten d to r a is e th e y ie ld p o in t , b u t t h i s i s

an in s u f f i c i e n t cause f o r work h a rd en in g .

This work must be con tinued w ith p o in ts o f much h ig h e r s t r a in b e ­

fo re a firm conclusion may be drawn concerning th e v a l id i t y o f th e th e o ry

th a t th e B auschinger e f f e c t i s caused by r e s id u a l s t r e s s e s .

45

Re c ommendati ons

h6

Since d i f f i c u l t y was encountered w ith th e Buda B-12 cement immed­

i a t e l y a f t e r y ie ld , i t i s recommended th a t th e u t i l i t y o f some o f th e h igh

e lo n g a tio n cements he in v e s t ig a te d . The Budd Co. produces a h igh e lo n g a­

t io n cement k i t d e s ig n a ted GA-1. Inc luded in t h i s k i t i s a p l a s t i c i s e r

which should s ig n i f i c a n t ly improve th e b ehav io r o f th e cement a t h igh

s t r a i n s . I t i s suggested th a t th e h igh e lo n g a tio n cement he t r i e d w ith

th e s tan d a rd epoxy gages a s w e ll as th e h igh e lo n g a tio n (p o s t y ie ld )

ty p e s . Perhaps t h i s method o f i n s t a l l a t i o n w i l l extend th e u se fu ln e ss o f

th e s tan d a rd g a g es .

When a paper gage i s cemented w ith Duco, th e gage and cement w i l l

u s u a lly s ta y i n ta c t a t e lo n g a tio n s as h ig h as 25,000 y, i n . / i n ^ I t would

he d e s ire a h le to check th e l i n e a r i t y o f th e se h ig h ly - s tr a in e d gages a g a in s t

some new ones. I f l i n e a r i t y s t i l l e x i s t s , i t may he p o ss ib le to use th e se

gages ag a in fo r th e com pression phase o f th e t e s t , th u s e lim in a tin g th e

need fo r i n s t a l l a t i o n o f new g a g es»

The in s t ru c t io n s f o r s t r e s s g rin d in g p r in te d by th e G eneral Motors

Corp. recommend u s in g a Norton type 32A^6-H8VBE g rin d in g wheel f o r t h i s

purpose . The Norton c a ta lo g may n o t l i s t a wheel o f t h i s type la rg e

enough to he used on th e a v a i la b le g r in d e r . However, an e q u iv a len t wheel

can e a s i ly he found. L arger d iam eter g rin d in g wheels a re recommended b e ­

cause o f th e le s s freq u e n t wheel d re s s in g n e ce ssa ry . O ther advantages

a re l i s t e d in C hapter I I I .

A ll th re e o f th e ty pes o f s t e e l t e s t e d showed th e B auschinger

e f f e c t . Any one o f them, th e 1018, 10^5 o r 11V* i s recommended fo r t h i s

type o f re s e a rc h .

^ T h is s t r a in i s w e ll w ith in th e work harden ing reg io n o f th e 1018s t e e l u sed .

b7

While th e lo ad in g o f th e specimens i s ta k in g p la c e , th e s t r e s s -

s t r a in diagram should he p lo t te d a s th e d a ta i s m easured. This procedure

w i l l enab le th e o p e ra to r to more c lo s e ly fo llow th e p ro g ress o f th e

t e s t . As th e d a ta i s b e ing m easured, th e t e s t in g machine should be op er­

a te d q u ite slow ly , bu t w ith o u t s to p p in g . When th e machine i s s topped ,

th e specimen ten d s to unload s l i g h t l y and in c o n s is ta n c ie s r e s u l t in th e

s t r e s s and s t r a i n d a ta .

D uring th e c u t t in g o f th e beams from th e specim ens, c a re fu l

m achining w ith l i g h t c u ts cannot be over-em phasized. In t h i s s tu d y , the

s id e s o f th e beams were ground in an a ttem p t to remove th e s t r e s s e s

caused by m achining.

When th e specimens a re s t r a in e d to th e v a rio u s p o in ts on th e

s t r e s s - s t r a i n diagram , i t would be advantageous to t r y pure com pression

on a few o f them fo r com parison p u rp o se s .

A curve f i t t i n g tech n iq u e u t i l i z i n g a method o th e r th an th a t o f

th e l e a s t squares should be t r i e d as an a ttem p t to smooth th e s t r e s s

p a t te rn s which a re g iven as o u tpu t by th e program . A nother p o s s ib i l i t y

would be to modify th e program so th a t th e curve f i t t i n g may be done

g ra p h ic a l ly by th e u se r so th a t th e f i t t e d curve may be m anually in p u t to

the computer

BIBLIOGRAPHY

1 . M artin , D. E. "E valuation o f Methods f o r Measurement o f R esidua lS t r e s s ," S o c ie ty o f Automotive Engineers TR-ll+7, September 1957.

2 . G u re rra , Carmin, "R esidual S tre s s G rin d in g ." L abora to ry Procedure,Used by th e ME-1 Department o f th e Research L ab o ra to rie s D iv i­s io n o f G eneral Motors C o rp o ra tio n . (V e rifa x e d .)

3 . Sm ith, S . L. and Wood, W. A. " In te rn a l S tre s s Caused by P la s t icFlow in M ild S te e l ," P roceedings o f th e Royal S o c ie ty o f London, l82:UoU-UlU, l p l T

*+. Wooley, R. L. "Bauschinger E f fe c t in Some Face-C entered and Body- C entered Cubic M eta ls ." P h ilo so p h ic a l Magazine, 1+1+:597-6 l8 ,Ju n e , 1953*

5* Naghdi, P. M., E ssenburg , F . , and K off, W. "An E xperim ental Study o f I n i t i a l and Subsequent Y ie ld S urfaces in P l a s t i c i t y , "Jo u rn a l o f A pplied M echanics, 25:201-209, June, 1958*

6 . C o t t r e l l , A. H. D is lo c a tio n s and P la s t ic Flow in C ry s ta ls (OxfordU n iv e rs ity P re s s , London, 1956).

7 . Handelman, G. H. and W arner, W. H. "Loading Paths and th e In c re ­m ental S tr a in Law," Jo u rn a l o f M athematics and P h y sics ,33:157-161+, 195^.

8 . Bridgeman, P. W. "The S tre s s D is tr ib u t io n a t th e Neck o f a TensionSpecim en," T ran sac tio n s o f th e American S o c ie ty f o r M eta ls ,3 2 :5 5 3 -5 7 U ,1 ^ i¥ :

9 . S hanley , F . R. "T en sile I n s t a b i l i t y (Necking) o f D u c tile M a te r ia ls ,"Aerospace E n g in eerin g , 20:30-31> December, 1961 .

10 . Nunes, John . "F lo w -S tre ss -S tra in R e la tio n sh ip in Tension T estso f S te e l ," M a te ria ls R esearch and S tan d a rd s , 3 No. 9 :719-722, Septem ber, 1963 .

11. P rag e r, W. "The Theory o f P l a s t i c i t y - A Study o f R ecent A chieve­m ents," Proceedings o f th e I n s t i t u t i o n o f M echanical E n g in eers ,169: I n - 57 , 1955.

12. V a sile v , A. M. "M icro s tre sses C reated in M etals During P la s t icD eform ation I I , " S o v ie t Physics o f S o lid S ta te , 1 , No. 11: 1586-1595, i 960 .

1+8

k9

13• H e ten y i, M. I . ( e d .) Handbook o f E xperim ental S tre s s A n a ly sis ,(John W iley and Sons, I n c . , New York, 1950),~ p . 5 l 8 .

14. Todd, Henry S. "R esidual S tre s s D is tr ib u t io n Changes During DynamicU n id ire c tio n a l T e n s ile Loading." Unpublished M as te r 's T h es is , Brigham Young U n iv e rs ity , Provo, U tah, I 96U.

15 . L e tn e r, H arold R. "A p p lica tio n o f O p tic a l In te r fe re n c e to th eStudy o f R esidua l S urface S tr e s s e s ," P roceedings o f th e S o c ie ty f o r E xperim ental S tre s s A n a ly s is , 10 No. 2 :2 3 -3 ^ , 1953*

APPENDIX A

EQUATIONS FOR RESIDUAL STRESS

The a lg e b ra ic ex p ress io n s fo r r e s id u a l s t r e s s a s d e riv ed by-

H arold R. L etner (15) have been review ed by th e a u th o r . The ab rid g ed

form i s p re sen te d a s fo llo w s: Let F igu re 21 re p re se n t th e c ro ss s e c tio n

o f a beam which i s caused to assume a c e r ta in c u rv a tu re , C, by i t s

r e s id u a l s t r e s s e s .

F ig . 2 1 .--Schem atic R ep resen ta tio n o f R esid u a l S tre s s

D is tr ib u t io n in an U n restra in ed B ar.

When th e beam i s c o n s tra in e d to l i e f l a t by th e a p p l ic a t io n o f

e x te rn a l moments, th e s t r e s s a t Z th en becomes cr' (z ) as shown in F igure

22. I f , w hile the b a r i s h e ld s t r a ig h t , s t r e s s e d su rfa ce la y e rs a re

removed by g r in d in g , th e le n g th o f th e beam w i l l change, g iv in g r i s e to

a uniform change a (w) in th e a x ia l s t r e s s th roughout th e th ic k n e ss of

th e rem ain ing m a te r ia l (see F igure 2 2b ). Now, th e s t r e s s a t any le v e l

z i s eq u al to th e o r ig in a l s t r e s s <j' (z ) p lu s a new s t r e s s a (w) r e s u l t ­

in g from th e change in le n g th (see F igu re 2 2 c ).

50

51

I f th e r e s t r a in in g e x te rn a l to rq u es a re re le a s e d th e specimen w i l l

bend to i t s e q u ilib riu m cu rv a tu re C(w) and in so doing w i l l s e t up an

a x ia l s t r e s s which can be deduced from F igure 21 . The a x ia l s t r a i n a t

le v e l z in th e b a r i s :

[R(w) + ( | - z ) ] &9 - R(w)d0 /W ^R(w)d0 = R(w) ^2 ” Z

where R(w) i s th e ra d iu s o f c u rv a tu re .

The correspond ing s t r e s s i s ob ta in ed as fo llo w s:

u s in g th e b a s ic beam equation

1 _ M_" R ‘ E l

M = CEI

Me= T

(8 )

(9)

( 10 )

s u b s t i tu t in g th e ex p ress io n s f o r M and c , th e moment re le a s e

s t r e s s i s o b ta in ed .

CT = C (w )E (| - z ) . (1 1 )

By adding th e o r ig in a l s t r e s s , th e le n g th change s t r e s s and th e

moment r e le a s e s t r e s s , t h e t o t a l s t r e s s i s o b ta in ed .

a (z ,w ) = a ' ( z ) + d(w) + C(w) E ( | - z ) . (12)

A pplying th e co n d itio n s o f s t a t i c e q u ilib riu m y ie ld s th e fo llo w ­

in g ex p ress io n fo r th e r e s id u a l s t r e s s which e x is te d a t any le v e l , z , b e ­

fo re any la y e r s were removed:

52

3 ( 3 , w0 ) = ^ ^ + ^ [ c (z ) - c (wo )]

w - z E r wo+ - y - E C(wQ) - j J z C(a)dz ( 13 )

where C (wq ) I s th e e q u ilib riu m c u rv a tu re b e fo re any la y e r s were removed

and C(z) th e e q u ilib riu m c u rv a tu re when th e beam th ic k n e ss had been r e ­

duced to w = z .

53

a. Jo.

Z Z

<S (Z,W) = s \ z ) + g(W) + C(W)E ( j - - z )

FIG. 22 .(a ). STRESS DISTRIBUTION WHEN THE BAR IS HELD STRAIGHT.(b ) . STRESS CAUSED BY THE CHANGE IN LENGTH RESULTING

FROM THE REMOVAL OF THE SURFACE LAYERS.

(c ) . STRESS DISTRIBUTION AFTER THE REMOVAL OF THESURFACE LAYERS BUT BEFORE THE BAR IS ALLOWED TO BEND.

(<J). STRESS CAUSED BY THE BENDING OF THE BAR TO ITS EQUILIBRIUM CURVATURE.

(e ) . STRESS DISTRIBUTION IN THE UNRESTRAINED BAR AFTER THE REMOVAL OF THE SURFACE LAYERS.

5U

FIG. 2 3 . GEOMETRICAL QUANTITIES NECESSARY FOR THE CALCULATION OF AXIAL STRAIN IN A BAR BENT TO A UNIFO RM C U R VA TU R E .

APPENDIX B

RESIDUAL STRESS GRINDING

By Cannin G u rre ra , Shop T echnician At Request o f J . 0 . Almen (2)

Forward

In r e s id u a l s t r e s s g rin d in g o f th e O.D. and I .D . o f a r in g or f l a t s u r fa c e s , th e co n d itio n o f th e wheel and depth o f c u t w i l l determ ine w hether s t r e s s e s w i l l be im parted onto th e su rfa ce o r n o t by g r in d in g .For s a t i s f a c to r y r e s u l t s , th e wheel must be k ep t c lea n and sh arp , (emphasis on sh a rp ) , and th e depth o f c u t must be d r a s t i c a l l y reduced from co n v en tio n a l g rin d in g p ra c t ic e fo r th e l a s t few thousand ths inch o f s to ck to be removed.

The fo llo w in g d e sc rib e s th e procedure o f s t r e s s g rin d in g as p ra c tic e d by th e ME-1 Department o f th e R esearch L ab o ra to rie s D iv is io n o f G eneral Motors C orpo ra tion fo r r e s id u a l s t r e s s a n a ly s is .

Procedure

F la t su rfa c e s (su rfa c e g r in d e r) :

Let us say , fo r example, th a t a .030 inch la y e r has to be ground o f f to some p redeterm ined dim ension. F i r s t , d re s s th e w heel. Proceed to remove th e f i r s t .020 inch in th e somewhat co n v en tio n a l manner o f .001 to .0015 inch depth o f c u t , depending upon th e s iz e o f th e work specim en. But a t no tim e s h a l l th e su rfa ce become d is c o lo re d o r o v er­h ea ted to such an e x te n t a s to make i t unbearab le to to u ch .

The rem aining .010 inch la y e r should be removed a s fo llo w s:

D ress w heel; remove .005 inch la y e r w ith c u ts .0003 inch deep ." " " .003 " " " " .0002 " "

" " " .002 " " " " .0001 "

For rem oval o f la y e r s o f .010 inch th ic k o r l e s s , proceed as o u tlin e d above.

The above schedule i s based on experim en ta l work done on s t e e l specimens o f 50 Roc hardness w ith a su rfa ce a re a o f approx im ate ly 2 sq . i n . (3A " x 3")* A f te r making f iv e com plete passes over th e specimen

55

56

w ith c u ts o f .0001 inch deep (.0005 inch t o t a l th ic k n e ss o f la y e r r e ­moved), th e wheel re q u ire d re d re s s in g . For p ro p o r t io n a l ly la r g e r a r e a s , say 10 s q . i n . (5" x 2" ) , th e wheel would re q u ire d re s s in g a f t e r one com plete pass w ith a c u t o f .0001 inch deep .

I t must he borne in mind th a t ex trem ely l i g h t c u ts w i l l ten d to d u l l th e wheel sooner th an h e a v ie r c u t s , th e r e f o r e , th e o p e ra to r must be v e ry observ ing in d e te c t in g prem ature d u l l in g o f th e w heel, g e n e ra lly seen as a b r ig h t o r g la r in g su rfa ce f i n i s h .

The ta b le t r a v e l r a te shou ld be medium, a lthough a f a s t e r r a te i s p re fe r re d over a slow er t r a v e l . The slow er r a te w i l l quicken wheel d u l ln e s s .

Cross feed o f th e work i s g e n e ra l ly from .015 to .020 inch p e r p a s s . For specimens .010 to .020 inch th ic k , i t re q u ire d g re a te r c a u tio n .

C onsecutive c u ts (down feed o f th e w heel) a re made a f t e r each f u l l coverage o f th e wheel over th e work i s made. No sp ark in g ou t i s n e ce ssa ry excep t f o r th e v e ry f i n a l c u t , i f d e s ire d , fo r a smoother f i n i s h .

G rind ing Wheel:

To d re ss th e w heel, c u ts o f .001 inch deep w ith a sharp diamond a re found to be most s a t i s f a c to r y .

The diamond should be ra p id ly passed under th e wheel in o rd e r to produce a f r e e c u t t in g w heel. A fte r making th e l a s t pass w ith a .001 inch depth c u t , do n o t rep ass th e diamond under th e wheel b e fo re c le a r in g th e wheel from th e diamond; o therw ise th e diamond w i l l on ly d u l l th e shaii) edges o f th e a b ra s iv e g ra in s on th e face o f th e w heel.

The p o in t o f th e diamond should be on c e n te r l in e o f th e wheel and th e n ib t i l t e d in th e same d i r e c t io n a s th e r o ta t io n o f th e w heel.The t i l t o r d rag ang le should be 10° - 15° from th e v e r t i c a l .

The wheel i s o f th e medium s o f t ty p e , w ith a v i t r i f i e d bonded No. k6 s iz e g ra in , and o f medium s t r u c tu r e . I t i s m anufactured by th e Norton Company and i s la b e le d 32A^6 -H8vBE. Symbol 32A s tan d s f o r Alundum a b ra s iv e ; No. k6 g ra in s iz e (medium); H grade ( s o f t ) ; No. 8 s t ru c tu re (medium sp ac in g ); V bond ( v i t r i f i e d ) ; and BE f o r type o f bond.

The w idth o f th e wheel i s l / k inch in s te a d o f th e con v en tio n a l l /2 " w idth to reduce d ra g . The range o f wheel speed i s 3300-3500 rpm.A ll work i s ground d ry .

O.D. G rind ing :

The same r a te o f s to ck rem oval as o u tlin e d f o r su rfa ce g rin d in g i s recommended, whenever p o ss ib le to ap p ly , depending upon th e s iz e o f th e work, r i g i d i t y o f th e work, and m achine. G enera l p ra c t ic e i s to p r a c t i c a l ly spark out b e fo re ap p ly in g a d d i t io n a l in - fe e d o f th e wheel to th e work.

57

Not much though t o r concern i s given as to w hether to g rin d wet o r d ry , s in ce th e depth o f th e cu ts a re ve ry l i g h t .

On th e two u n iv e rs a l g rin d in g machines used fo r s t r e s s g r in d ­in g , one employs a l /2 " x 12" wheel a t 1700-1800 rpm and th e o th e r has a l /2 " x 10" wheel running a t 2600-2700 rpm.

Again on ly about 1 fk" w idth (by d re s s in g ) o f th e •wheel i s used fo r both m achines. The grade o f both wheels i s p r a c t i c a l ly th e same as fo r th e su rfa ce g r in d e r , 32AM$-l8VBE.

I .D . G rind ing :

E x ac tly th e same recommendations and s p e c if ic a t io n s as th o se fo r O.D. g r in d in g , except th a t th e wheel speeds used a re as recommended by th e m anufactu rer.

The depth o f c u ts w i l l v a ry w ith th e s iz e o f work, r i g i d i t y o f th e q u i l l , and m achine, e t c . , in v a r ia b ly le s s th an th ose s p e c if ie d fo r su rfa ce g rin d in g .

58

APPENDIX C

OUTLINE OF THE STRESS CALCULATION FRCGRAM

1 . Read in th e t i t l e , re g u la r p o in t d a ta .

2 . Compute th e c o rre c te d cu rv a tu re fo r r e g u la r d a ta .

3 . Read in th e in v e r te d d a ta .

b. Compute th e c o rre c te d c u rv a tu re fo r th e in v e r te d d a ta .

5 . Compute th e c u t t in g d is ta n c e .

6 . P r in t ou t th e c a lc u la te d d a ta .

7 . Read in th e modulus o f e l a s t i c i t y end in s t ru c t io n s fo r th e

p lo t te d cu rve .

8 . I n i t i a l i z e th e curve f i t t i n g eq u a tio n c o u n te r .

9 . Read in th e no . o f eq u a tio n o rders used and eq u a tio n o rd e rs .

10. Compute th e curve f i t t i n g m a tr ix .

11. In v e r t th e computed m a trix .

12. W rite out th e computed fu n c tio n s .

13. Compute th e d is ta n c e and f i t t e d c u rv a tu re e q u a tio n s .

lU . W rite out th e computed fu n c tio n s .

15. C alcu la te th e mean s t r e s s f o r th e second cu t s t r e s s com putation.

16. Compute th e o r ig in a l s t r e s s v e rsu s th e d is ta n c e below th e specimen

s u r f a c e .

17. W rite out th e computed fu n c tio n s .

APPENDIX D

A LISTING OF THE MODIFIED STRESS

CALCULATION PROGRAM

n n

r>

n n

n

n n

n

no

n

no

n n

o

C R E S I D U A L S T R E S S DATA A N A L Y S I S 0 0 1 0C 0 0 2 0

D I ME NS I ON T ( 3 0 ) * 0 ( 3 0 ) » G ( I O . I O ) . 0 0 3 0I C C R V E C 5 C ) . C C U R V E ( 5 0 ) • C L R V E K S O ) , C I S T ( 5 i > , S ( 5 0 ) . 0 1 ( 1 2 ) . 0 0 4 02 i H I C K ( 5 0 ) 0 0 5 0

C DI ME N S I ON T ( 3 0 ) . D ( 3 0 ) . X ( 3 0 ) . H ( 1 0 ) , G ( I O . I O ) , G I V < 2 0 . 2 0 > . A ( 2 0 ) . 0 0 6 01C U R V E ( 5 0 ) * C C U R V E i S C ) . C L R V E K S O ) , C I S T ( 5 C ) » S ( 5 0 ) » D I U 2 ) , C( 3 0 ) , 0 0 7 02 T H I C K ( 5 0 ) 0 0 8 0

CCMMCN/ SAMEVGI V ( 2 0 , 2 0 ) , H ( 2 0 ) » MP1 . N P , C ( 3 0 ) . X ( 3 0 ) . A ( 2 0 )

READ DATA I C E N I F I C A T I C N CARD AND D I S E C T I N CUT NUMBER

5 0 READ < 5 . 1 0 ( D M I ) . I - 1 . 1 2 )I C FORMAT ( 1 2 A6 >

READ ( 5 . 1 4 ) NCL T « CC 1 4 FORMAT ( 15 / ( E 1 5 . C ) )

READ I H I C KN E S S AND CURVATURE DI S T ANCE - OUT S I DE OF S P ECI MEN UP

READ ( 6 . 1 1 ) N RP » ( T ( l > , D ( I ) . I = 1 . N R P )I I FORMAT ( 15 / ( 1 2 F 6 . 0 ) )

COMPUTE CORRECTED CURVATURE

DC I 0 0 1 = 1 , N R PICC C < I > 2 . 0 * D ( I ) + CC / 1 ( 1 ) * « 2

READ N O . OF DATA PCI NT 5 TAKEN WITH S P ECI MEN I NVERTED

REAC < 5 . 1 2 ) M P 1 2 FORMA 1 ( 1 5 )

NP = NRP + N I P N R P 1 = NRP + 1

READ DATA P O I N T S TAKEN WI TH S PECI MEN I NVERTED

0 0 9 0 0 1 0 0 ouo 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0160 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0 2 7 0 0 2 8 0 0 2 9 0 0 3 0 0 0 3 1 0 0 3 2 0 0 3 3 0 0 3 4 0 0 3 5 0 OnO

no

n

no

n

no

n

no

n

no

n

nn

READ (5*13) C T ( I ) « D(I), I = NRP1, NP ) 036013 FOR MAT ( 12F6.0 ) 0370

C 0380COMPOTE CORRECTED CURVATURE 0390

04C0DO 110 I = NRP1i NP 0410

110 C (I ) = 2.0* D ( I ) - CC / T { I ) «* 2 04200430

COMPUTATION OF CUTTING CISTANCE 04400450

DO 120 I = 1, NP 0460120 X (I ) = TCI) - TCI) 0470

0480PRINT OUT CALCULATED DATA 0490

0500fcRITE(6*1) NCUT * CC 0502

1 FORMAT C/ 2 9 X 6HNCUT = 13* 5X 4HCC = E15.5 /) 0503fcR I TE (6*15)01 * ( I, TCI), XCI), DCI), CCI). 1=1,NP ) 0510

15 FORMAT!// 29X 12A6 /// 21X 1HI 18X 4HTCI) 16X 4HXCI) 16X 4HDCI) 0520116 X 4 H C ( I )//( 17 X 15. 5 X 4F20.8 )) 0530

0540READ IN MODULUS OF ELASTICITY, NO. OF PLOTING POINTS AND INTERVAL 0550

0560READ (5,16)N P P , XINT, E 0570

16 FORMAT! I5.2F15.0 ) 05800590

READ ORIGINAL SPECIMEN THICKNESS VALUE 06000610

READ ( 5 ,13)CRIGT 06200630

INITIALIZE CURVE FITTING ECUAT ION COUNTER 06400650

ICOUNT = 1 0660C 0670 o,C READ NO. OF EG. ORDERS USED AND EQUATION ORDER 0680 HC 0690

r, n

REAC ( 5 . 1 2 I N M 1 3 C READ ( E . 1 2 ) N

MPl = M + 1CC COMPUTATI ON CF CURVE F I T T I N G MATRIX

DO 1 3 2 J = 1 . MPl1 3 1 H { J ) = 0 . 0

DC 1 3 2 K = 1 . MP11 3 2 G ( J » K ) w o * c

DO 1 3 7 J = i » MPlDO 1 3 7 I = l . NPIF(J-l) 135. 133. 135

133 H{J) = H (J ) + C ( I )GO TO 1351

135 H (J ) = H (J ) + C (I ) * X(I) ** (J-l)1351 CONTINUE

DO 137 K = 1, MPl IF(J + K— 2) 136. 1353. 136

1353 G (J »K ) = G(J,K) + 1.GO TO 137

136 G(J,K> = G(J.K) + X(I) ** (J+K-2)137 CONTINUE

DC 140 J = 1 ,MPl A(J) = H(J)DC 140 K = I .MPl

1 4 C GIV(J.K) = G(J.K)

INVERT COMPUTED MATRIX

DO 150 J = 1. MP1WRITE (6,18)J , H (J ) , ( G(J,K),

18 F O R M A T (/1 OX 3HJ =. 12.( 11E 12.4) )15 C CONTINUE

CALL MAT INV

0 7 0 00 7 1 00 7 2 00 7 3 00 7 4 00 7 5 00 7 6 00 7 7 0 0 7 8 0 0 7 9 0 0 8 0 0 0 8 1 0 0 8 1 2 0 8 1 4 0 8 1 6 0 8 2 0 0 8 2 5 0 8 3 0 0 8 3 2 0 8 3 4 0 8 3 6 0 8 4 0 0 8 5 0 0 8 6 0 0 8 7 0 0 8 8 0 0 8 9 0 0 9 0 0 0 9 1 0 0 9 2 01 000

K = l , MP 1 ) 1 0 1 01 0201 0 3 0 CJ\ro0 9 3 00 9 4 0

no

n

o

Ccccc

3 5 CC

WR I TE OUT C C MPU TED FUNCTI ONS

COMPUTATI ON CF DI S T A N C E AND F I T T E D CURVATURE F UNCTI ONS

DO 3 5 0 1 = 1 , NPP C U R V E ! I ) = 0 . 0 D C U R V E ( I ) = 0 . 0 CUR VE 1 ( 1 ) = 0 . 0D I S T ( I ) = T H I C K ( I ) = DO 3 5 0 J = CUR V E ( I ) = D C U R V E ( I ) = C U R V E I ( I ) = CONTI NUE

FLCA T ( I ) CRI GT - 1 , MP1 C U R V E ( I )

DCURVE(

* X I NT D 1 S T ( I )

+ I )

CURVE I ( I )

A ( J ) * DI S T ( I ) * * { J - l )+ F L O A T ( J —1 ) * A ( J ) » D 1 S T ( I ) **+ A ( J ) * D 1 S T ( I ) ** J / F L OA T ( J )

( J - 2 )

WRITE OUT COMPUTED FUNCTI ONS

WRITE ( 6 , 2 0 ) 0 1 , N P P , M, Z , E , C R I G T , ( I , D I S T ( I ) , THII C K ( I ) , D C U R V E ( I ) , CUR V E ( I ) • C U R V E I ( I ) , 1 = 1 , N P P )

2 0 FORMAT( / / 1 0 X 1 2 A 6 , / / / 1 0 X 6 HNPP = , 1 4 , 10X 4HM = , 1 4 , 10X1 3HZ = E 1 5 . 6 . 1 0 X 3HE = E 1 2 . 5 , 10X 7HCRI GT = F 1 0 . 7 , / / 1 0 X 1HI2 1 1 X 7 H D I S T ( I ) 1 2 X 8 H T H I C K ( I ) 1 1 X 9HDCURVE ( I ) 1 2 X 8 H C U R V E ( I ) 1 1 X3 9 HCURVE 1 ( 1 ) / ( / 8 X 1 3 . 5 F 2 0 . 8 ) )

I F ( NCUT - 1 ) 9 9 8 . 4 0 0 , 4 5 09 9 8 WRITE ( 6 . 2 1 )

2 1 F OR MA T ! 1 0 X 4 3 HCAL L E D E X I T AT 9 9 8 DUE TO ERROR CONDI TI ON )CALL E XI T

CCALCULATI ON CF MEAN S T R E S S FOR 2ND CUT S T R E S S COMPUTATI ON

4 0 0 CF I N A L CF I N I DO 4 1 0 C F I N A L

= 0.0 = 0 . 0 J = 1 , MP1= C F I N A L + A ( J ) * X( NP) ** ( J - I )

0 9 5 0 0 9 6 0 1 1 8 0 1 1 9 0 I 200 1 2 1 0 1 2 2 0 1 2 3 0 1 2 4 0 1 2 5 0 1 2 6 0 1 2 7 0 1 2 8 0 1 2 9 0 1 3 0 0 1 3 1 0 1 3 2 0 1 3 3 0 1 3 4 0 I 3 5 0 1 3 6 0 1 3 7 0 1 3 8 0 1 3 9 0 1 4 0 0 1 4 1 0 1 4 2 0 1 4 3 0 1 4 4 0 1 4 5 0 1 4 6 0 1 4 7 0 1 4 8 0 1 4 8 2 1 4 8 4 1 4 8 6

CNLO

u u

4 1 0 CF I N I - C F I M + A ( J ) * X ( N P ) * * J / F L O A T ( J ) 1 4 8 8CHCLD = C F I N A L 1 4 9 0OTMI ST = —( T ( N P ) * CFI NAL + C F I M ) « E / 3 . 0 l 5 0 0QHOLD = OTMI ST l 5 1 QWRITE < 6 , 2 2 > D I . C F I N A L . CT V 1 S T . C F I M l 5 2 0

2 2 FORMAT! / / 2 9 X 1 2 A 6 , / / 2 9 X 8 HC F I N A L = , E 1 5 . 6 , 10X 8 HQTMI S T = , i 5 3 01 E 1 5 . 6 , 1 0 X 7 H C F I N I = . E 1 5 . 6 ) 1 5 4 0

CF I N A L — 0 • C 1 5 5 0OTMI ST = O. C l 5 6 06 0 TO 5 C 0 1 5 7 0

4 5 C I F ( I COLNT - 1 > 9 9 7 . 4 5 2 . 4 5 4 1 5 8 09 9 7 WR I TE ( 6 . 9 7 ) l 5 g 0

9 7 FORMAT( 10X 3 1 HCALLED E X I T AT 9 9 7 CUE TO ERROR ) 1 6 0 0CALL E X I T 1 6 1 0

4 5 2 READ ( 5 , 4 5 3 ) C F I N A L , OTMI ST 1 6 2 o4 5 3 F O R M A T ( 2 E 1 5 . 0 ) 1 6 3 04 5 4 I F ( C F I N A L ) 5 C 0 » 4 6 C » 5 0 0 1 6 4 04 6 0 CF I N A L = CHCLD 1 6 5 0

OTMI ST = QHCLD 1 6 6 0C 1 6 7 0C ORI GI N A L S T R E S S V S . DI S T ANCE BELOW S PECI MEN SURFACE COMPUTATI ON 1 6 8 0C 1 6 9 0

5 C 0 DO 5 5 0 1 = 1 . NPP 1 7 0 05 5 0 S ( I ) = ( ( T ( 1 ) — D I S T ( I ) ) * * 2 * D C U R V E ( I ) / 2 . 0 - 2 . 0 * ( T ( l > - D I S 1 7 1 0

1 T ( I ) ) * C U R V E ( I ) + CURVE 1 ( 1 ) ) * E / 3 . 0 - D I S T ( I ) * E* C F I N A L —QTMI S 1 7 2 02 T 1 7 3 0

C 1 7 4 0WRITE OUT COMPUTED F U NCT I ONS 1 7 5 0

1 7 6 0WRITE ( 6 . 2 3 ) D I , CHOLD, CHCLD. C F I N A L , OT MI S T , ( I , O I S T ( I ) , 1 7 7 0

1 T H I C K ( I ) , S ( I ) . 1 = 1 , NPP ) 1 7 8 02 3 F O R M A T ! / / 2 5 X 1 2 A 6 , / / 2 1 X 7HCHCLD = E 1 5 . 6 . 5X 7 HQH0LD = E 1 5 . 6 , 5X 1 7 9 0

1 8 HC F I N A L = E 1 5 . 6 . S X 6 H C T M I S T = E 1 5 . 6 . / / 12X I H I 1 0 X 7 H D I S T ( I ) 1 3 X 1 8 0 02 8 H T H I C K ( I ) 1 2 X 4 H S ( I ) / ( / 1 0 X 1 3 . 3 F 2 0 . 8 ) ) 1 8 1 0

I F ( NM - I CCUNT ) 5 6 0 , 5 0 . 6 0 0 1 8 2 05 6 0 WRITE ( 6 » 2 4 ) NM » I CCUNT 1 8 3 0

n n

2 4 FORMAT ( 1CX 4HNK = , 1 5 .1 5 5 0 . >

CALL EXI T

8H I COUNT = , 1 5 . 5X 2 1 H E R R 0 R CONDI TI ON AT 1 8 4 0 1 8 5 0 1 8 6 0 1 8 7 0 I 8 8 0 1 8 9 0

6 0 0 I CCLNT = I CCLNT + 1GO TC 1 3 0 END

$ I B F T C MATI NV L I S T C MATRIX S OLUTI ON CF L I N E A R EQUATI ONS

1 SUBROUTI NE MATI NVM A 0 0 2 0M A 0 0 4 0

2 COMMON/ S AME/ A( 2 0 . 2 0 ) . B ( 2 0 ) . N . K . Y ( 3 0 ) . X ( 3 0 ) . C ( 2 0 )3 M1=N4 M2 =N+15 D 0 6 1 = 1 . Ml6 A ( I , M 2 ) = B ( I )

DC 4 0 J = 1 . Ml4 0 1 k R I T E ( 6 . 4 1 ) ( A ( J , I ) . 1 = 1 , M 2 )4 1 FORMAT( 1 H + . 1 0 F 1 3 . 5 )

*R I TE ( 6 , 4 2 ) N . M 1 . M2 . K . M4 2 FORMAT ( 1 H + . 5 I 6 )5 2 DO 6 0 1 1 = 1 . Ml53 L 1=1+15 4 DO 6 0 J = L 1 . M25 5 SOL = A ( I , J ) / A ( I , I )5 6 I F ( S O L ) 5 8 . 6 0 , 5 85 8 DO 5 9 L = J , M 2

A t J , L ) = A ( J . L ) - A ( I , L ) * S C L5 9 W R I T E ( 6 . 4 3 ) I . J . A ( J . L ) • S O L • A ( I , L )4 3 FORMAT! 1 5 X . I 3 , I 3 . 3 F 1 5 . 3 )6C CONTI NUE

6 0 1 CONTI NUE6 1 DO 6 8 1 = 1 , Ml6 2 L 3 = M 2 —I6 3 S L M = C • 06 4 I F { 1 - 1 ) 6 8 , 6 8 , 6 5

6 5 L 2 = F l - I + 26 6 DO 6 7 L = L 2 * FI6 7 S U M = S U F 4 C ( L ) * A ( L 3 , L >

* R I T E ( 6 , 4 3 ) L 3 * L » SUF *C ( L ) » A ( L3 » L )6 8 C ( L 3 ) = ( A ( L 3 , F 2 ) - S U F ) / A ( L 3 . L 3 )

C CO ERROR ANC VALUE CCFP1 6 8 S U F = C . C

6 9 VkRI T E ( 6 , 7 0 ) (C ( I ) , 1 = 1 , M7 0 F C R F A T ( 5 X , 1 8 H C C E F . OF CURVE F I T / 5 X , 1 OE1 2 . 7 )7 1 CO 7 9 1 = 1 , K7 2 F E X = 0 . 07 3 DC 7 5 J = I » N7 4 L = N + 2 —J7 5 F E X = ( F E X + C ( L ) ) * X ( I )7 6 F E X = F E X + C ( 1 )7 7 ERR = Y ( I ) —FE X7 8 S UF= S UM+ ERR* ERR7 9 V n R I T E ( 6 . 1 3 0 ) X ( I ) , Y ( I ) . F E X . E R R

13C F O R F A T ( 6 X » 2 H X . 9 X , 1 0 H F E A S . C U R V . , 7 X , 1 OHCALC, CURV. , 4 X , 5 HERR0R 1 / 4 E 1 5 . 7 )

3 2 6 DE V = 13 2 7 D E V = ( S Q R T { S U F ) / D E V )3 2 8 V k R I T E ( 6 » 1 1 8 ) D E V1 1 8 F 0 R F A T ( 1 H + , 1 0 X , E 1 1 , 5 )

8 3 RETURN 8 1 END

SENTRY

85

APPENDIX E

SAMPLE INPUT SHEET AND EXPLANATION

SAMPLE OF I NP LT DATA 1

1 2 . 5 E — 71 3

. 1 5 1 . 0 0 0 5 . 1 3 4 . 0 0 0 0 . 1 3 1 - . 0 0 0 5 . 1 2 0

. 0 8 6 - . 0 0 3 0 . 0 7 6 - . 0 0 2 0 . 0 6 5 - . 0 0 6 0 . 0 5 4

. 0 3 0 - . 0 3 0 03

. 0 5 4 - . 0 1 0 . 0 4 4 - . 0 1 8 0 . 0 3 0 - . 0 3 12 4 C . 0 0 5 0 2 9 5 C 0 0 0 0 •

. 1 5 114

P OI NT N O . 1 CONTROLS SECOND SAMPLE1

o • o

13. 1 9 4 0 . 0 0 0 0 . 1 8 3 8 . 0 0 0 0 . 1 6 7 7 - . 0 0 0 1 . 1 4 9 9 -. 1 0 1 3 - . 0 0 0 7 . 0 8 5 2 - . 0 0 0 9 . 0 6 9 6 - . 0 0 1 4 . 0 4 8 6 - . 0 2 8 3 - . 0 0 5 9

C

3 4 C . 0 0 5. 1 9 4 0

1 3 234

2 9 5 0 0 0 0 0

NCUTCCNRP

. 0 0 0 0 . 1 0 9 . 0 0 0 0 . 0 9 8 - . 0 0 0 5 T . D 1 - 6

. 0 1 0 0 . 0 4 4 - . 0 1 7 5 . 0 3 5 - . 0 2 1 0 T . D 7 - 1 2T . D 1 3N I PD . T 1 4 - 1 6 NPP I NTE OR IGT NM = 1

M = 4

CUT = 1 CC = 0 NRP

. 0 0 0 2 . 1 3 3 0 - . 0 0 0 3 . 1 1 7 6 - . 0 0 0 5

. 0 0 2 2 . 0 3 8 0 - . 0 0 3 3 . 0 3 3 6 - . 0 0 3 8

N I P = 0 BLANCK NPP I NTE OR IGT NM=i NM=3 M=2 M = 3 M=4

8

APPENDIX F

DATA ORDER AND FORMAT DESCRIPTION

A d e s c r ip t io n o f th e symbols used in th e program as re c e iv e d

from Henry S. Todd ( l^ ) i s given as fo llo w s:

Data I d e n t i f i c a t io n Card See Symbol Table D ire c to ry

Specimen Cut Number P lace f ix e d p o in t number 1 o r two in column 5 (Format 15)

C o rrec tio n C o e ff ic ie n t I f you tak e measurements o f d e f le c ­t io n on bo th s id e s o f th e specimen du rin g th e g rin d in g procedure t h i s c o e f f ic ie n t i s unnecessary . Use zero i f t h i s i s th e c a se . I f n o t , 12 .5 tim es 10"? i s p robab ly a s good as any . (See sample d a ta . )

NRP E n te r h e re (Format 15) th e number o f d a ta p o in ts to be used in th e s e c ­t io n th a t fo llo w s . I f CC = 0 , I would su g g est th a t a l l your d a ta be p u t in t h i s s e c tio n s im ila r to specimen l 6 SP, c u t 1 on sample d a ta s h e e t .

Thickness and D e fle c tio n D istance Here i s where th e Thickness andd e f le c t io n d a ta go. Each card w i l l

NIP

handle 72 c h a ra c te rs in th e form of 12 s ix c h a ra c te r f i e l d s . P lace specimen th ic k n e ss f i r s t and th en i t s measured d e f le c t io n from th e cu rv a tu re guage.

I f you use a zero c o rre c t io n co ­e f f i c i e n t (CC = 0 ) , p lace a zero a ls o in column 5 h e re and p lace a b lank card a f t e r t h i s one.

Data Taken in in v e r te d p o s it io n

NPP INTE This ca rd co n ta in s th re e f i e ld s w ith th re e b i t s o f in fo rm a tio n .

69

NPP INTE

TO

F i r s t , (NPP) i s th e number o f r e s u l ta n t c a lc u la te d p o in ts to be p lo t te d . N ext, (INT) i s th e i n t e r ­v a l used between each c a lc u la te d s t r e s s and d e f le c t io n v a lu e . L a s t ly , (E) i s th e modulus o f e l a s t i c i t y .The card f i e ld s a re : 1 s t , a f iv e c h a ra c te r f ix e d p o in t f i e l d fo r NPP, and 2nd, two 15 c h a ra c te r f lo a t in g p o in t f i e ld s fo r INT and E. (ORIGT - NPP*INT should equal th ic k n e ss o f specimen a t tim e change from 1s t c u t to second i s made. )

ORIGT O rig in a l o r f i r s t value o f th ic k n e ss to be used in p lo t t in g th e r e s u l t s a t o u tp u t. (F 6 .0)

NM See Symbol D ire c to ry (Format 15)

M See Symbol D ire c to ry (Format 15) I f NM is la r g e r than one, th e program w i l l look fo r a d d i t io n a l curve f i t t i n g o rd e r cards a f t e r t h i s on e . The c a lc u la t io n s made on th e second c u t d a ta uses cu rv a tu re d a ta from th e f i r s t c u t . I t i s th e re fo re im­p o r ta n t th a t th e curve f i t t i n g o rder t h a t you f e e l i s b e s t be p laced l a s t i f more th an one i s to be u sed . There i s a p ro v is io n b u i l t in to th e program whereby second c u t inform a­t io n can be p rocessed in d ependen tly o f f i r s t c u t d a ta , b u t I h a v n 't found occasion to use i t y e t .

CFI QTM CFI s tan d s fo r CFINAL and QTM re p re ­se n ts Q sub t minus S sub t . I t i s w ith t h i s card th a t p ro v is io n fo r p ro cess in g 2nd c u t d a ta independ­e n t ly o f f i r s t c u t d a ta i s p rovided f o r . D iis card must appear a f t e r th e f i r s t curve f i t t i n g o rd e r (m) ca rd in each 2nd c u t d a ta s e t . I f m u ltip le curve f i t t i n g o rders a re u sed , th e r e s t fo llo w t h i s c a rd .In p ro cess in g in th e s tan d a rd manner where f i n a l cu rv a tu re va lues from th e f i r s t c u t a re used w ith th e se c ­ond, p lace f lo a t in g p o in t zeros in each o f th e se 15 c h a ra c te r f i e l d s .

G eneral Note 1: Sign Convention

For both f i r s t and second c u t , when th e s id e o f th e specimen where th e g rin d in g i s c u r r e n t ly ta k in g p lace i s p laced a g a in s t th e su rfa ce o f th e cu rv a tu re guage, a p o s i t iv e c u rv a tu re and d e f le c t io n d is ta n c e e x is t s i f th e c e n te r o f th e specimen i s lo w e st. (Same shape as p o s i t iv e moments). For th e f i r s t c u t th e n , th e o u ts id e o f th e specimen would be th e to p as th e specim en 's c u rv a tu re was b e in g measured A f te r th e d e f le c t io n d is ta n c e was measured in th e s tan d a rd manner as d e sc rib e d above, i t shou ld be tu rn e d over and read a g a in , t h i s value b e ing m u lt ip l ie d by -1 s in ce i t was in th e in v e r te d p o s i t io n d u ring measurem ent. By u sin g bo th o f th e se v a lu e s , a c o r re c t io n w i l l a u to ­m a tic a l ly tak e p lace in th e computer curve f i t fo r th e e f f e c t o f th e w eight o f th e ends o f th e specim en.

G eneral Note 2:

L ast va lue o f d e f le c t io n f o r f i r s t c u t = 1 s t fo r 2nd c u t . The f i r s t v a lues fo r d e f le c t io n d is ta n c e and specimen th ic k n e ss fo r th e 2nd s e r ie s o f c u ts i s th e l a s t v a lu es f o r th e 1 s t s e r ie s o f c u ts in to th e specim en.

SYMBOL TABLE DIRECTORY - RESIDUAL STRESS DATA ANALYSIS BY H. TODD

cc C o rrec tio n C onstant (Used to c o r re c t specimen cu rv a tu re due to w t. o f ends o f specim en)

T Thickness o f Specimen

D D istance c e n te r o f specimen i s below th e l in e connecting to p o f two inch c e n te rs on c u rv a tu re m easuring in s tru m e n t.

NIP Number o r d a ta p o in ts tak en w ith specimen in in v e r te d p o s i t io n .

NRP Number o f d a ta p o in ts taken w ith o u ts id e o f specimen up which i s re g u la r p o s i t io n .

DI Data I d e n t i f i c a t io n card (can c o n ta in 72 c h a ra c te rs o f any alpha-num eric in fo rm atio n )

NP T o ta l Number o f P o in ts (NP = NIP + NRP)

NRP1 NRP1 = NRP + 1

C C urvature o f Specimen

X D istance o f Cut

NCUT Cut Number (Has value o f e i th e r 1 o r 2 depending o f w hether d a ta i s from 1 s t o f 2nd c u t o f specim en).

NPP Number o f p o in ts to be used in p lo t t in g th e r e s u l t

XINT

E

CURV

THICK

M

NM

H

Z

DIST

DCURVE

CURVEI

CFINAL

ORIGT

72

I n te r v a l between p lo t t in g th ic k n e sse s

Modulus o f E la s t ic i ty -

Curve f i t t e d C urvature

Curve f i t t e d Specimen Thickness

Order o f Curve F i t t i n g Polynom ial

Number o f Curve F i t t i n g Polynom ials to be used

Summation Location (See F o r tra n Program)

A Measure o f th e accu racy o f th e curve f i t to th e o r ig in a l d a ta

Cut D istance used in p lo t t in g f i t t e d cu rv a tu re v s . X

dC/dX%CdX

CURVE (NPP)

O rig in a l Thickness o f Specimen

RESIDUAL STRESSES AND THE

BAUSCHINGER EFFECT

An A b stra c t o f a T hesis

Subm itted to th e M echanical E ng ineering Department

Brigham Young U n iv e rs ity

Provo, Utah

In P a r t i a l F u lf i l lm e n t

Of th e Requirem ents f o r th e Degree o f

M aster o f Science

■by

Raymond M. Huebner

Ju n e , 1965

ABSTRACT

The prim ary o b je c tiv e was to f in d th e r e s id u a l s t r e s s p a t te rn in

a p l a s t i c a l l y deformed b a r in an a ttem p t to r e l a t e t h i s s t r e s s p a t te rn

to th e B auschinger e f f e c t . S te e l specimens were p l a s t i c a l l y deformed in

te n s io n and com pression and then analyzed fo r r e s id u a l s t r e s s by the

beam d is s e c t io n method. P re lim in ary work was done to e s ta b l i s h r e s id u a l

s t r e s s g rin d in g tech n iques f o r fu tu re re s e a rc h .

The problem o f bending f i e ld s in th e t e n s i l e t e s t i n g machines was

c a r e f u l ly in v e s t ig a te d and c o r re c t iv e m easures were ta k e n . A p iece o f

ap p a ra tu s u t i l i z i n g tw in b a l l jo in t s was designed to minimize th e bending

d u rin g com pression and th e lo c i o f th e axes o f average d e f le c t io n were

c a lc u la te d f o r a l l o f th e s i tu a t io n s encoun tered .

A specimen c o n fig u ra tio n was designed which would ex p ed ite th e s t r e s s

g rin d in g procedure and f u l f i l l th e o th e r requ irem en ts o f th e s tu d y . Lab­

o ra to ry te ch n iq u es , i . e . , s t r a i n gage i n s t a l l a t i o n , s t r a i n m easurem ents,

and g rin d in g procedures were in v e s t ig a te d and d is c u s s e d . V a ria tio n s in

s t e e l type and machines used were t r i e d in a search f o r advantageous com­

b in a tio n s .

I t was found th a t y ie ld in g f i r s t occu rred in th e outerm ost f ib e r s

o f bo th th e t e n s i l e and com pressive specim ens. The s t r e s s p a t te rn s

found sup p o rt th e id ea t h a t th e B auschinger e f f e c t cou ld be caused by

r e s id u a l s t r e s s e s . No d e f in i t e co nclusions were reached because o f th e

l im ite d amount o f d a ta m easured and because o f s c a t t e r in th e c a lc u la te d

s t r e s s p a t t e r n s .

Improved methods fo r s tu d y in g t h i s problem a re recommended.

1

APPROVED