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NASA CR-165563

{_AZA-C_-1655o3) }ATIGUE LI_E _PhED1C_.[CN iN N_Z-2-05o_bENDiNG _SOM A_AL _A_IGU_ IB_UBNATIC& (taseWeste[a Rese[ve Oz_lv.) __ p ticAO3/_ AOJ

CSCi 2OK Ul_ci_G3/J9 09385

FTIGUE LIFE PREDICTION IN BENDING

FROM AXIAL FATIGUE INFORMATION

._ S.S. MansonProfessor,Mechanical and Aerospace Engineering

and

U. MuralidharanResearch Assistant

i

Case Western Reserve UniversityCleveland, Ohio 44106

February 1982

1982012690

https://ntrs.nasa.gov/search.jsp?R=19820012690 2020-05-23T09:28:39+00:00Z

NASA CR-165563

%

FATIGUE LIFE PREDICTION IN BENDING

FROM AXIAL FATIGUE INFORMATION

S. S. Manson

Professor,Mechanical and Aerospace Engineering

and

U. MuralidharanResearch Assistant

Case Western Reserve UniversityCleveland, Ohio 44106

February 1982

1982012690-002

INTRODUCTION

It is common to express the bending fatigue life of a material in

terms of the nominal elastic stress or strain calculated by assuming

stress is proportionalto strain. Yet, in the low-cycle fatigue range,

and in fact over a wide range of life of engineering Interest, plastic

flow is involved,and stress is not proportional to strain. Thus con-

siderablediscrepancy exists between bending fatigue and axial fatigue

when the latter involves expression in terms of true stresses and strains.

While several factors contribute to this discrepancy, as discussed in

Ref. l, among them the presence of strain gradient, a volumetric effect,

cyclic strain hardening and softening effects, and crack propagation

differences after the surface stress has been initiated, the major effect

is due to the fact that the nominal stress and strains on a specimen in

bending differ considerably from the true values due to plastic

deformation.

In Ref. l, it was demonstrated that flexural fati£Je is consider-

ably different from axial fatigue, but that the two could be brought intoi

J coincidencewhen based on true surface stresses. Fig l shows some of|

the results of the reference. The solid curve and circular data points

shou the axial fatigue characteristicof 4130 steel. In this case the

stresses are true values because in axial loading it is simple to deter-

mine the stress from a knowledgeof load and cross-sectionalarea. The

dashed curve and square data points refer to the beam bending

1982012690-003

characteristics. Here the stress is calculated from the conventional

McS - I assumptions,so that the stresses are the fictitious values

determined on the basis that no plastic flow occurs. It is seen that

for this material the nominal bending stress can be 501%higher at the

lO0 cycle life level, and that significantdifference persists between

the two curves until well above lO5 cycles to failure. Obviously, then,

it is importantto reconcile the differences over a wide range of life

levels of practical engineering interest. It wFs also demonstrated in

Ref. l, that if the true stress is calculated in the case of the bending

specimen the li_e comparisQn with the axially Ioa_ed comparison is much

more favorable. But, because of the difficulty of performing the inte-

grations involved in the bending case no closed-form solution was

presented. Each material and geometry had to be treated numerically.

It is the purpose of thi_ report to extend the work of Ref. I and

to obtain closed-form generalized relationships. The approach is first

to study rectangular rather than circular sections. Not only are

rectangularsections of great importance in engineering applications,

but they result in relations that can be integrated in closed-form.

The resulting form of the solutions then provides a model for use in

the case of circular sections. By assuming an analogous model,

involving several undeterminedconstants, and then determining the

constants so as to fit a range of known solutions obtained by numerical

integration,it has been possible to obtain very accurate, though not

exact, closed-form solutions for the case of the circular section.

The procedure and results will be described in this report.

$

I

1982012690-004

3

In addition to the development of plasticity as a reason for _difference between axial and bending fatigue, another factor relating

to volume under stress also introduces differences in the lives of the

two loading modes. In pure axicl loading the entire cross section of

the specimen is subjected to the loading stress, whereas in pure bend-

ing only the surface fibers are subjected to the maximum stress, while

other regions of the cross section are subjected to lower stresses.

The effect of this factor is expected to be important in the high cycie

f_,tiguerange, rather than the low cycle fatigue range which is of major

interest in this report. However, this subject is also briefly

discussed. _

BASIC EQUATIONS USED FOR LIFE RELATIONSHIPS

The Conventional Life Relationship

The basic model for axial fatigue used in this report is shown in

Fig. 2. Although the model was first proposed by Manson (Ref. 2)

using a different notation, the notation used in Fig. 2 is the one that

has been commonly adopted ( Ref. 3). The strain amplitude, -_ is

plotted on the vertical axis, and the number of reversals 2Nf on the

horizontal axis, using logarithmicscales on both axes. If a material

is axially cycled at constant strain range about a zero mean value, or

if the plastic strainrange is maintained constant, it is generally

found that in the early cycles the stress (or, equivalently,the elastic

strain) may change considerably, but eventually a saturation value is

1982012690-005

4

achieved. Usually this saturation value develops early and persists _

for the major part of the life of the specimen. Plots such as shown in

Fig. 2, in which both the plastic strain amplitude, and the elastic

strain amplitude, are plotted against reversals to failure result in

straight lines PQ and LM, the equations of which are, respectively,

P = E_(2Nf)C (I)2

A_e _ Ao _ o_ )b2 2E E (2Nf (2)

where A_p is the plastic strainrange,Ace is the elastic strainrange,!

and Ao the saturation stress range, _f and T are the intercepts

at 2Nf = l of the plastic and elastic lines, E is the elastic modulus, and

c and b are the slopes of the two lines as shown in Fig. 2.

The equation for the total strain amplitude, which is the sum of the

plastic and elastic components, is thus

AE cf(2Nf)C + o_2 - ' -E-(2Nf)b (3)

In Ref. 4 we introducedthe term "transition life" to designate the

life at the intersection point T of the elastic and plastic lines. In

Fig. 2 this life is denoted NT and the number of reversals is 2NT.

Correspondingly,the strainrange at this point is A_T and the strain

amplitude is A_T/2. By equating the strains _n Eqs. (1) and (2) the

values of NT and A_T can easily be determined.

L

1982012690-006

b-cI _f

t

b c

: _Jf c-bA_T = 2 E -_-

] n (5)

= 2 c

!

(

i ci where n = g . It may be noted that n is the reciprocal of the strain'+ I

" hardening exponent m' in the relation Ao : Constant (Acp) m . From

Eqs. (4) and (5) it can readily be shown that the basic life model

Eq. (3) can be rewritten [5l

_ : + (_)'_T

The Inverted Life Relation

For convenience in some of the numerical integration analyses we

have also used an alternative form of the life relation discussed in

Ref. 6. Basically, life is expressed directly in terms of strain,

instead of using Eq. (6) which expresses strain in terms of life

1z z 1

I N_f_f= [ _ b z (7)NT Re + RtIwhere

ii

(i

i!

i

1982012690-007

6

[ ( 36)]z = exp P In2 RE _ .+Q In R +In - 889c ( ) (8)

P = -.001277 (_) + .03893 (_) - .0927 (9)

2

Q = +.004176 (_) - .135 (_) + .2309 (lO)

Ac

RE = AC---T--ratio of strain required to produce life Nf tothe transition strain (ll)

For any material of known b, c, NT and AcT , all the constants

involved in the ee, :tions are readily calculated, the life Nf is ex-

pressed directly in terms of those constants and strainrange.

Similarly, stress range An can be written directly in terms of :

applied strainrange AE

The Universal Slopes Life Equatlon: In cases where material

constants b, c, E'f and o'f are not accurately known from experiment,

it may be desired to estimate the life relation on the basis of common-

ly available material properties. In Ref. 1 it was demonstrated that

as a first approximation c could be taken as a universal value of -0.6,

b as a universal value of -G.12 and that _' could be estimated from, f

the ductility, while o'f was best estimated from ultimate tensile

strength. The resulting "universal-slopes" equation, as expressed in

Ref. (I)= is*,

-0.6 3.5ou(13)Nf-0.12A_ : D0"6 Nf + E

t *While this equation is generally credited only to Manson because

, it first appeared in Ref. l, the equa_ contribution of M. H. Hirschbergin its development is acknowledged.

1982012690-008

where

D = ductility = -In (I-RA) wherein RA is reduction of area in _tensile test

o = ultimate tensile strength,and E is elastic modulus inu consistent units.

Expressed with the SAE notation of Eq. (3), the equation becomes

(2Nf)-O. °uAE2- 0"7578D0"6 6 + 1.902 T (2Nf)-0"12 14)

= 7578D0"6 = 15)That is, c'f O. and o'f 1.902ou

Also, using Eqs. (4) and (5), _,

_u 1.25 15AcT = 4.789 (T) D-" (16)

o -2.083, u_ D1.25

NT : .0735 _-i (17)

so that Eq. (6) becomes

- + aT (18)

where AcT and NT are given by Eqs. (16) and (17), respectively.

Similarly, substitutingc = -0.6, b = -0.12 into Eqs. (9) and

(lO), we obtain P = .07, Q = -.3397

z : exp[.07 In2Rc -.3397 InRE -1.208]

R = AwLc a_T

Nf : NT [RLz/c + RcZ/b] TM (19)

where A_T, NT are given by Eqs. (16) and (17), respectively.

1982012690-009

FL_XURAL BENDING OF RECTANGULAR CROSS-SECTIONS

We first treat the bending of rectangularcross-sections because _

the resulting equations can be integrated in closed form. As discussed

in Ref. l, it is more convenient to solve this type of problem in an

inverse fashion compared to conventional treatment. Rather than start-

ing with a known bending moment, and determining the resulting surface

strain--whichwould involve solution of non-linear equations.-it has been

found better to select a surface strain and calculate the uending momen_

required to produce this strain. From the selected surface strain we

establish the fatigue life, and from the required bending moment the

nominal elastic stress. Thus we can e_tablish the relation between nominal

elastic stress and fatigue life.

Fig. 3 shows the notation used in the analysis. We assume a sur-

face strain Ls correspondingto a life NS when the full bending moment M

is applied. Since in bending it is co_m_n to assume that plane sections

remain plane, strain varies linearly across the thickness, so that the

strain _ at a distance y from the neutral axis (center of bar) isY

Ey = _ _s (20)

Substitutingfor _s in terms of life NS associated with the surface

strain, from Eq. (6)*

_TJ[Ns b (Ns)C ] (21)Cy = h lINT + NT"

*Becaus--e-ofthe symmetry er loading for this ¢_se of completelyreversed bending, we omit the a's, and consider only surface strainsat the maximum loading condition.

1982012690-010

Now if Ny is the life associated with ttcestrain at y, then by Eq. (6)

Y = Ny + t2ZET NT NT

and the stress associated with the strain 6 is, as derived from Eqs.Y

(2), (4) and (5), "

b

Y NT :

The increment of bending moment contributed by a strip w dy at tne

distance y from the n_utral axis becomes

• -- INylb. wy dy (24) ,dM = y Oy w dy E cT_NT)

and the total bending moment, taking into account the symmetrical contri-

bution to bending moment of the section below the neutral axis is

y=h

M = dM = 2Ewe y dy (25)

Here Ny is a function of y according to Eqs. (21) and (22). In order

to carry out the integration it is best to establish dy in terms ofi

N N

__ZNT, rather than expressing --_in terms of y, which would result in a

more difficult integration. Thus from Eq. (21) and (22),

NT + NTy : h--- (26)

NS )b NS

1982012690-011

10

and ",_

,N ,c1]_y: ?" b +cl_T] d (_7)(_s){5?NTT + _NT]

From Eq. (25), therefore,

NS

_w__, I/__'o-_j_/u'c- _c _I)'NT/_NTIJ i

NS

2Ewh2£T F'N""' IN' 2b+C-1 (_TT; b+C-1 (_ D+2C-] (:_,NS + NS ' /

(29)

NS

( b ,.12 NT * 2-b_c * _ NT

NS_ /NsX'I

N

In Eqs. (28), (Zg) and (30) the lower limit for N-_Tis taken atI

I_= 0, where the strain is zero, and therefore the life is infinite.

1982012690-012

1

Since all the exponents of NT in the integrated expression are ne(jative, 'i_the value of all terms is zero at the lower limit, and M is evaluated (J_

+ 2-b+-c b+2c _ NTM = 2Ew h2ET (31)

Ns\2b b+c NS\ 2c

The nominal surface stress Sbending is Mh _ 12Mh = 12M since theI w(2h)3 8wh2

total height of the rectangle is 2h and the _ment of inertia of a

rectangularsection is h base x (height)3. Thus, substituting for _

fron:Eq. (31), and dividing both numerator and den_inator by NT '

the resulting equation for Sbending is obtained as follows"

INs) c-b 3c IN_S_2(c-b)

Sbending = EET/NS-_b_N-T) I�23-'b+Lb-_ NTT + _\NT/Ns 2(_bT--

b

(NN_T)SaxiaI = EtT (33)

SbeDdi__= correction factor • 6 (34)Saxial

(c-b) 2(c-h)NS

1 * 2 _T'T/ ,NT/

1982012690-013

12

where

fl - 3(b+c) _ 1 + n _2b+c (2/3) + (1/3)n where n = c/b (36)

3c 3nf2 - b+2c - l+2n (37)

FLEXURAL BENDING OF CIRCULAR CROSS-SECTIONS

We calculate the bending moment required co prod_me a selected

surface strain. From the selected surface strain fatigue life can be

esLablishedand similarlynominal stresses from the required bending

moment. Thus we establish the relation between nominal elastic stresses

and fatigue life.

Figure 4 shows the notation used in the analysis. We assume a

surface strain cs correspondingto a life NS when the full bending

moment [_is applied. Strain varies linearly across the section, so that

the strain c at a distance y from the neutral axis isY

= Y-EsLy R

y = R sinO , dy =RcosOdO (38)

Ey cs sinE) (39)

_nd the stress associated with the strain Ey can be found using Eq. (12),

= I(Es sin E))z/b _s sin E))z/clb/zOy EET CT + • _._- (40)

1982012690-014

13

where z is given by Eqs. (8) to (ll). _

The increment of bending moment contributedby a strip at the

distance y from the neutral axis becomes

dM -- y. Oy-(width of the strip)-dy

Substitutingfor y and dy from Eq. (38) and o from Eq. (40),Y

[(E sino)Zlb ( sin{_izlclZlb -dM = R sinO. EcT s + .Es 2Rcos_ RcosOdc :

J t_

(41)

The total bending moment, taking into account the symmetrical contribu-

tion to bending moment of the section below the neutral axis is

=RM = 2 dM

y=O

_/2 [ICsSinoi/b +I_s sinoi/Cl z/bM = 4R3ELT _ CT _ ET I ] sin O cos2__d,,"- (42)0=0

From Eq. (6),

where z is a function of (c/b) and (Cs/_T) given by Eqs. (8_-(II).

It is clear from Eq. (42) that the integral does not lend itself to

simple closed form solution. However, for any oiven material and a given

surface strain, the integral can be computed numerically and the moment

can be calculated. The nominal surface stress Sbending is expressed as:

1982012690-015

14

M R M R 64 64 M

Sbending - I - _R4 _R3

The ratio of nominal surface stress S to the stress that would be

required to produce a life Ns in axial fatigue, can be written analogous

to Eq. (35) by assuming the denominator and the exponent (c-b) remain

the same for both rectangular and circular cross-sections.

(NsI (c-b)+ f2/__S_S_2(c-b)

where fl and f2 are for the circular cross-section. The ratio 6 is calcu-

lated numerically using Eqs. (43) and (42) for different values and

combinations of parameters involved as follows:

NS _ 10-3 ' 10-2,...,102 , 103NT

c = -0.3, 0.4,...,-0.9,-I.0

b = -.05,-.06,...,-.19,-.20

Equation (44) can be written as

[ 11 + 2 _TT + _TT) -- 1+ fI_TT] 2\NTJ (45a)

:, (N s )(c-b)

1982012690-016

15

( )G NTT' c,b,6 = fl + f2 H NTT' c,b (45)I

If the function G is plotted against the function H for a given

NS

(c/b), i.e. n, but for all combinationsof c,b, and NT , a straight

line results. Thus the slope f2 and intercept fl are functions of only n.

The plot of G vs. H for different n values are shown in Fig. 5. For clarit,,

the numerical values of functions G and H for different combinations of

c, b and Ns/NT are shown in Fig. 5 only for the case of n= 5. The above

procedure is repeated for several materials of various n values and the

parameters fi and f2 are calculated for each material with a specific n

They are shown in Fig. 6.

Since the parameters fl and f2 are found to be functions of only n,

an attempt was made to determine these functions whose form is analogous i

to ones shown for rectangularcross-section in Eqs. (36) and (37).

The functions were determined by least squares analysis to be

fl - l + .8432 n (46).6849+.255n

2.6188 n (47)f2 = i+I.5411 n

The curves represented by the above functions are shown by solid lines

in Figure 6.

For any given material the functions fl and f2 can be calculated using

the reciprocalof the strain hardening exponent, n, and Eqs. (46) and (47).

The correction factor 6 and axial stress SaxiaI can be calculated for any

given life using Eqs. (44) and (33), respectively. Thus we can plot nomi-

nal bending stress Sbending against life. The plots are shown in Fig. 7

1982012690-017

16

for several materials of engineering interest. The properties of these _materials,obtained from Ref. 7, are listed in Table II.

If the fatigue properties of the material are not known or cannot

be obtained easily, Universal Slopes solutions can be used with the

properties obtained from a tensile test. Universal plastic and elastic

ex_onents of -0.6 and -0.12 are used, respectively. Transition strain

\_.T and transition life NT can be determined from Eqs. (16) and (17),

resF_ctively.

DISCUSSION

The approach presented in this report takes into account only the

effect of plastic flow in explaining the difference between axial and

bending fatigue lives; hence, it is valid in the low cyclic life range.

According to the analysis, the SaxiaI and Sbending curves become coinci-

dent for most engineering materials at lives greater than lO6 cycles

_ince the strains are purely elastic at thoselife levels and the Mc/I

assumption is valid. Howevur, bending fatigue life could still be

higher than at axial fatigue for the same nominal stress due to volumetric

effect which is not taken into account in the current approach. This is

observed by Esin in steels and he has used micro-plastic s,,ain energy

criterion to take this factor into account (Ref. 8). This is also

disclosed in some detail in connectionwith risk.of-rupture criterion

;nRef. 9. Basically, in bending fatigue lower volume of material is

under high strain at outer fibers as compared to the whole volume inC

the axial case. Thus, the smaller the volume under maximum stress,

"! ORIGINAL PAGE ISt OF POOR QUALITY

i

1982012690-018

ORiG;_',,iALPAGE IS

OF POOR QUALITY17

the lower will be the probability of the presence of a fl_w at a point

of high stress, hence the higher will be the life. _

At high cyclic life range, the rotating bending will give lower life

as compared to alternating bending for the same nominal stress which can

e explained as follows. In alternating bending the regions A and B _n

Fig. 4 will be subjected to maximum surface strain, whereas in rotatin;

bending all the material near the surface of the cylinder wiJl be sub_c¢

to maximum strain at different times; for example, the regioH C will taket

the position of A or B while rotating, hence will be subjected to maximum

strain. Thus, mnre volume is subjected to high stress in rotating benViqg

hence greater will be the probability of the presence of a flaw at a point _

of high stress and lower will be the life.

It should be noted that results obtained in this report are valid

for alternating flexural bending. If the material is subjected to rotating

bending, the cross-sectiondeflects about a different axis, which makes an

angle with the loading axis ( Ref. l). Once the angle between two axes it

determined, the complete stress and strain distribution can be computed

from which the bending moment for an assumed value of maximum surface strain

can be established. In Ref. l i_ was observed that although the analysis

for rotating bending is considerably more complicated than static flexuralc

bending, the resulting strain distributions for the two cases differea

little when considered in their respective bending planes. It was shown

that the experimental results for rotating bending agreed well with the

predictionsmade for static flexural bending. Hence it will be assumed here

that the equations for flexural bending, Eq. (44) together with the fl and f2

values given in Eqs. (46) and (47), are also valid for rotating bending In

: the low cycle fatigue range. Further study may be desirable to verify thi_I

i assumption.

Ii

I!

1982012690-019

18

CONCLUSION

Closed-form expressionswere obtained for nominal bending stress

in terms of axial fatigue stress based on true surface stresses. They

are shown for rectangularand circular cross-sections in Table I. Using

only axial fatigue data on a given material and formulas given in Table I,

nolnlnalbending stress-life curve can be established. Though the study

is !innted to only rectangular and circular cross-sections,the approach

and the procedure described in this report can be applied to more compli-

cated geometries to obtain the functional forms for parameters fl and f2

which are functions of only strain hardening exponent of the material.

The approach takes into account only the effect due to plastic flow which

has dominating effect in the low cyclic life range. However, in the

high cyclic life range other factors could be more dominating like the

volumetric effect which should be taken into account in reconciling the

difference between axial and bending fatigue lives. F_:turestudy is

necessary to incorporate the volumetric effect in the analyses presented

in this report.

1982012690-020

REFERENCES _

I. Manson, S.S., "Fatigue: A Complex Subject - Some Simple Approximatio_."Proceedings Society of ExperimentalStress Analysis 12, No. 2 (1965_

2. Manson, S.S., "Thermal Stress in Design, Part 19, Cyclic Life of Ducti_"Materials," Machine Design (July 7, 1960), pp. 139-144.

3. Morrow, J., "Cyclic Plastic Strain Energy and Fatigue of Metals,"ASTM STP 378 (1965), pp. 45-87.

4. Smith, R.W., Hirschberg, M.H., and Manson, S.S., "Fatigue Behavior of"_ Materials Under Strain Cycling in Low and IntermediateLife Range,

NASA-TN-D-1574, April 1963.

5. Manson, S.S., "Predictive Analysis of Metal Fatigue in the High CycleLife Range," Methods for Predicting Material Life in Fatigue, The

Winter Annual Meeting of the ASME Dec. 2-7, 1979, pp. 145-183. _.

6. Manson, S.S. and U. Muralidharan, "A Single Expression Formula forInvertingStrain-Life and Stress-Strain Relationships,"NASA CR-165347.

7. Landgraf, R.N., Mitchell, M.R. and LaPointe, N.R., "Monotonic and CyclicProperties of Engineering Materials," Ford Motor Company (1972).

8. Esin, A., "A method for correlating different types of fatigue curve,"InternationalJournal of Fatigue, Vol. 2, No. 4, Oct. 1980, pp. 153-158

9. Manson, S.S., "Thermal Stress and Low-cycle Fatigue," Robert E. Krei_£rPublishing Company, Malabar, Florida, 198 , sec. 7.1.7 and 7.1.8.

1982012690-021

-, q

20

TABLE I

n : c/b ratio of plastic to elastic exponents Nk

_ = transition strain range = 2of']q-6

!

where cf = fatigue ductilitj coefficientI

of -- fatigue strength coefficient

E = modulus of elasticityl

_' -- Transition life = 1 _'T (Tf'I

b

, = ECT< NS)axi.,l _TT

- (c-b) 2(c-b)-

NS

Sbendi ng : I+2 ( NS_(c-b')_--. +( Ns _2(cL'b)NT/ Saxi elL "T- _

where fl and f2 are functions of n and depend on the geometry of the

cross-section.

For Rectangular Section For Circular Cross-section

l + n fl l + .8432 nfl : (2/3)+ (I/3)n : .6849;.255 n

3n f2 2.6188 nf2 : _ : I+1.5411 n

For any given surface strain Es' surface life Nr,,can be found froP Fig. 2 drd

SaxiaI and Sbending can be computed using the above formulas.

1982012690-022

_ 1

luDl_ , 1i

t ', , i It ,

NATIJI IM. a. el b C F '_t T 1 _ *

kit| kltl t CYClee

__ _. | .......

r#I L_IG_)_-IO_ IIILLC 11 .1| -.07') - _I 2_0(03 2 tellj ]K_)O,3

12 _ILIM[I (NRLC) 117 _ -.071 -.6S 2q)200 4 151t_ _,]7_

-w

_t a,tXlGt_-ltl$ magi 170 1,0 -.0q_ -.&6 290C_3 *_ ?::,, _ tll_

_i|} ttmld045--lq)O IU4N _]tO ,45 -.0?t -.68 ICY@O@ g )1I_,_ 6_6

#6 1_11045-410 IINll 270 .6_) -.O't) - 70 lq(*oo .011'_6 18._

1982012690-023

2_ ORIGINAL PAGe i_

OF POOR QUALITY.Table II (Continued)

_. ,,-em,mm_

i *

IMgMII_& • t 61 b C [ t_T _ *bi •at C]_: |cO

emqp...-,..m,m_ .....

#2i 10| It_pf11414NI •N 'J2_ .4_ -.01 .15 _kOQO ,017 ,Sdl

I]IO Zl_IG-_-T_ 12) .62 *.|(_ .6_ I O(YJO .0|)4 3]'9

/)1 _'Ql6-* I - T4 1&7 .2| -.Jl ._2 |0200 .OJSO'_ ),kk

e)z _4_6 A:--t--- :o_ ._, -.;, ..67 loooe _.99xi_ •_7

.l0)t Ir,4g95Oi-l$O IJlOI 91 .15 -.075 .56 IoGt3Q 2.JIt2ll(, lib.Or

|)_ V&1140--22_ _ 1_3 .ll °.Of ._] 28200 _.6711( |b_6)

-_,#]W_ 81_100-294J i 147 ._ -.076 ._,l 294,00 _.t2ll(I 1_12

#)1 ItU106 5..-5,00 0ira )]W) .25 -.04 .t4 looon .014'_ 91

#)4 AIIItl)_-2M I ll_ .92 -.OO) .b) ]2000 _,)6ll( _/98

_I9 K4,JA162*]tSO J484 _6_, .6_ -.08 .75 ]K)043_ t ,OllO' 17_

I•O 1,4J_l•l--_O Illll )0_ .&O -.09 .76 29000 : .0122 209

,_llO0_161 I;ti_WO.-)_O _ 260 .13 -.O_b .hi' 't. Tlxl( 1/61

#tl AISI_iOO-_|8 Id_ )I_ .18 -.09 ._4 JW)O0_ .OI') l•b

#4) SN[gItI-_f80 _ 117 .61 -.073 ,kO ]JllO00 l ).0911( J)_]'I

d_6 l-El kSO _ 660 .0_ -.071 .76 )OQQ_ ,025) 6

I_,_ tlsl]u(_-)21 I_x )1o I1_ -.17 .•,u ?5000 010,_ Ioe

1_6 ,MI)_O-•IN_ONle )90 .044 -.102 .t2 2t,OO0 , OlS,, 18'_

#•1 ill Ilckll _r_sll_ |60 .) -,045 .•2 21000 Oll_ _046SO IW_

2O]6*Tl}t t t uatmm IM_ .2] I - . l _ll6 i - . $¢:[ | 04100 . Ol_tO ie)l

_$ lOll-Y• admtmm 191 ,19 -.lit -.'_2 10)oo .oJ_, |l* (

#_0 S&g,lll_$-MIkl_ 9) .I -.IC# -.)9 ,_lO00 i •gIILI 10)¢,_J

1982012690-024

23t

I

200;- \

o. Rotating beambending

g 120 -

_n

s,.

4O I I , A I J101 102 103 104 105 I(/"

Cyclesto failure,Nf

Fig. I. Fatiguedata underaxial loadingand in rotatingbendingfor4130 steelfromRef. 1.

1982012690-025

I

Fig, 2. Model of fatigue life relationshipconsisting oflinear elasLic and plastic components on doublelogarithmiccoordinates.

1982012690-026

2fl

T,,+h j[

Fig. 3. Rectangularcross-sectionalbar in flexural bending.

Fig. 4. Circular cross-sectionalbar in flexural bending.

1982012690-027

26

_w

0 I , ! ! ....... I

0 25 50 75 100

(' )H NT ,c,b

Fig.5. Plotof G vs. H fora givenc/b and for allcombinationsof c, b and (N_/NT)showingfl and f2a-e functionsof only (c/b)._ "

1982012690-028

27

3.1 - 0 SAE I015 - 80 BHN _-0- VAN BO - 225 BHN

RQC lO0 - 298 BHND SAE I045 - 500 BHN

3.0 -_ SAE 4142 - 380 BHN

AISI 4340 - 409 BHND AISI 52100 - 518 BHN

4_ H-ll - 660 BHN

2.9 - _ SAE 9262 - 280 BHNAISI 304 - 327 BHN

-_- 18% Ni Maraging 460 BHN61 = Eq. (46)

' 2.8 - _ AM 350 - 496 BHN

2.7 - - l7

fl = Eq. (47)

2.6- - 1.6 ,_

2.5- - 1.50 2024 T 351 Al

-£_ Manten - 150 BHN f2

2.4- _) AISI 4130 - 258 BHN - 1.4

m SAE 950X - 150 BHN

• SAE 4142 - 450 BHN2.3i- - 1.3

I2.31 ' I , i , I , i , l ,

1 3 5 7 9 II 3

n

Fig, 6. The parameters fl and f_ are shown for materials withdifferent n values. Th_ solid lines are curve-fitsgiven by Eqs. (46) and (47), respectively.

1982012690-029

ORIGINALPAGE tS28 OF POOR QUALITY

0, , _ o 01 _? I03 10_ lOS 10b lC lu" 103 IC _ Ic _

Ltfe, k I cycles Life, NI cvcle_

, _\. l,oI "_\ 1111 Ul Klta|lm& 460 Ig_

_120

:'c, _ .._ s , _ ; Sb,'_ds_

12'_ 60

ol ;- ....L_|e. N1 cv_leb LI_, _ ,,,_q

15G ]

S._ 950X, l_t_ IJ_

._ Sbendin _

_ 5mj, j

0 O'

1 _ 1(? 103 |0 _ l 0_ lO II _0 102 ) 01 1C'_ , C_ 1C

I, ll*, _l tyc|e_ L_Ie. kf cyc_e_

Fig. 7. Axial stress and nominal bending stress are plotted againstlifefor variousmaterialsusing formulassummarizedin Table I.

1982012690-030

29 ORIGINAL PAGE i3

OF POOR QUALITY.

O_ iO2 lO ) |0 & 10 '_ 106 10 107 |0 j :._ 10"

Llfe I|. ¢y¢|e_ Lilt NI . c_l_

211(1, 250

12,

O* 01 ................

IG I( _ IO ) I0 _ IG': |0(' 10 II1" I(. _ I0" I: _ 1''Ll|e

_O(, __1 }4 _ AISI $280_ big' IN',

I,,0 150

1_,_ tO'_ _0 ) I@_k |0 $ |0 li 50 10 ! |0 ! lO_' iO $ i,.,_

I.llo Ill, Cyclll l.llo I_, Cy(io,

Fig, 7 (Continued)

1982012690-031

30 O_IG_N..'J. ' - "

L _ 9262. 280 8_,

r_lng S_'nd lni_

'C' 10" 1 lO_" IO S _06 1O 102 If ' ,_

,,_c. Life NI , Cycit_ _$0 _,_e b c_ ,*

AIS1 JOd. 117 iNN $bendll,_

_ Sbe_dIn|

iI

O* ' , 0 I -

22" LAf* _f. CvLle_ JT_ l_ft _I _''''"

|_c_ _0_ T)_J A_nuB _0 ]St N_ Nsraql_q 40', BHN

oi oI0 102 I03 1[." 105 10 t 10 101 lO 1 IC_4 I0_ IOb

Life Nf. Cycle, l, lfe, I;f, Cycle|

Fig. 7 (Continued)

1982012690-032

, . ,, , i , i

jo 10 2 103 Io + I0 s 10 + 10 I0 _ 10 ] 1o I 10 _ I_+.

LAte, Mt, Cycle8 L_fe, Nf, C_ycles

3?5 110 LSI3ondxng IA.I[: 1005 - lO0) HRL_

_+'g IJUt 1045 - 410

125 l0 1

0 . , , , 0 -,_0 102 10 ) IG 4 |0 S 106 10 102 10 3 l04 10 S 3{

L_te, Nt, Cycle| L_{m, _f, C_clms

ll04M__n_ 3O0

GAXI_X (NIU,C)

; 12o I00

• +_ g

60 IO0

0 1 l i Ol l_ _ ,.. i.

lO lO + 10) 10 4 lO S lO t lO 101 10_ 10 4 10 + it

life, I l, Cycles &+re, mr. Cycles

Fig. 7 (Continued)

1982012690-033

3? ORIGff'JAL P,_,,3_J,_OF POOR QU;_.LIT'Y

lJG _ SA.1£ _26_ - _60 iNN ]7_I_L SAZ 4J42 - 4_', BH_

i

[Ip

_ i," Io z 1o4 lo _ io_ Io 1o 2 Io_ _4 - -._,

"' _Jfr. Nf. L'ycles )00 Lafe. _f, Cycles

$_ 4_42 - 400 BH_, _ AISI 4J4C - 24J B_

_L _$i_ .200

riding

'_ 10o

cl o ;_3 I02 iO ] 104 lO _ 106 IO 102 i0" _C4 _

S_F.. 4142 5A[ _E_ - 43C Bh_

_ °

d g

Io Io2 |03 104 10_ I0 & I0 I02 Ig _ 104 I0"

_e, Nf, Cy¢le_ L_f¢, hr. C_clt_

Fig. 7 (Continued)

1982012690-034

33

ORIGINALPAGE IS

OFPOORQUALITY

0 0

1o 1o _ 103 ]04 105 106 10 102 103 l_ 4 _*

_75 LItI, k . Cyc|es 4SO_Life. hf, Ct¢;e_

S_nd_'n_l AIS| 4).}0 " 365 lINN S_E 4_.4,_ - 4"_ SH_,

i,°°0

Io lo 2 I0 } :_4 I0$ i0 t 10 10 2 l_ ) 104 l_'

L_f_, hf, C)_it_Life, NR. Cycles

4SO 600,

sb..d,. _ _z *1,2 - s*oi.. s...__, T_-t_,-_.

i :r, g

l_O _rO0

0 , * Ol

10 10 2 10 ] 10 4 IO S 10 i 10 10 2 IG 3 ;0 4 IC;

L_fe. Jll, Cycles L;fe. t,f. Cycles

Fig.7 (Continued)

1982012690-035

34

Gt" POOR QU;.Lki'_

i SAL 9_0_ - 4_,0 IHk :)GI4 A_ T(_

0_c 102 IQ 3 |04 1oS l06 l0 l02 _0 j l(_4 .c _

45_ LHe, It|. Cycles 37S1 LH*. _f, C.w./es :

10t NI _facpm,j 410 BO4F, /Dend_l,_;

,B0 |IS

el, 0l0 1°" l01 1(,4 _0 _ 10t lfJ 10 _ lO 3 l° 4 ic '_

es_ LHe, k I , Cycles 4_vl ;.;|e, kf, Cyc;es4

0¢

10 lO'7 10) $040 IO $ ;O t IO 10 _ ;0 _ 10 4 i0_ ,

J_ie, I1|. C'yC|mS Ls|e, _t' cycles

Fig.7 (Continued)

)

),

1982012690-036

!

1982012690-037