UNCLASSIFIED

fiepAoduced

¡Hf UlA

ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA

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NOTICE: When government or other drawings, speci¬ fications or other data are used for any purpose other than in connection with a definitely related government procurement operation, the U. S. Government thereby incurs no responsibility, nor any obligation whatsoever; and the fact that the Govern¬ ment may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or other¬ wise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use or sell any patented invention that may in any way be related thereto.

£7

23

9^0

ADD TECHNICAL REPORT 61-177

4

«a" STRUCTURAL SAFETY UNDER CONDITIONS OF ULTIMATE

I LOAD FAILURE AND FATIGUE

w~*1

A. M. FREUDENTHAL

M. SH1NOZVKA

COLUMBIA UNIVERSITY

I

*»

4E3l GO

OCTOBER 1%1 I A i rn r“

1%2

r-f—n r^\

ili SSA

ÊtyJ 3

AERONAUTICAL SYSTEMS DIVISION

WADD TECHNICAL REPORT 61-177

STRUCTURAL SAFETY UNDER CONDITIONS OF ULTIMATE

LOAD FAILURE AND FATIGUE

A. M. FREUDENTHAL M. SHIN07Ä'KA

COLUMBIA UNIVERSITY

OCTOBER mi

DIRECTORATE OF MATERIALS AM) PROCESSES

CONTRACT No. AF 33(616)-7042

PROJECT No. 7351

AERONAUTICAL SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND

UNITED STATES AIR FORCE

WRICHT-PATTERSON AIR FORCE BASE, OHIO

700 - January 1962 - 16-694 & 695

FOREWORD

This report was prepared by the Department of Civil Engineering and Engineering Mechanics of Columbia University, under USAF Contract Mo. AF 33(616)-7042. The contract was initiated under Project No. 7351, "Metallic Materials", Task No. 73521, "Behavior of Metals". The work was administered under the direction of the Directorate of Materials and Processes, Deputy for Technology, Aeronautical Systems Division, with Mr. D. M. Forney, Jr. acting as project engineer.

This report covers the period of work 1 February I960 to 31 November I960,

The cooperation and continued interest of Mr. D. M. Forney, Jr. is gratefully acknowledged.

ABSTRACT

The purpose of this investigation is to analyze the concept of the safety of structures subject to operational loads that cause fatigue damage as well as to occasional excessive overloads that might produce ultimate load failure.

In Part I the relation between probability of failure and the reliability or i; safety iactor is discussed. Diagrams have been computed under the assumptions that the statistical variations of load and carrying capacity are expressed either by log-normal or by extremal distributions. The safety of multiple load-path structures, the probability of failure of simple structures under combined (primary and secondary) loads are also considered and the use of separate load factors for dead and live load is related to the concept of a single safety factor.

Part II deals mainly with the statistical properties of fatigue life distri- butions. Assuming a statistical-mechanical model for the fatigue mechanism, a new distribution of fatigue lives is derived. The concept of stress-interaction established in previous experimental research is used to reproduce the survivorship functions under random loading from the known survivorship functions associated with constant stress amplitude fatigue.

In Part III the risks of ultimate load and fatigue failures are combined and the reliability of aluminum specimens (AA 2024 Al) under both operational loads and occasional excessive overloads is investigated considering the interrelation with the risk-functions. The procedure is illustrated by a numerical example in which the truncated part of an exponential load spectrum is applied as operational (fatigue) loading while the rest of the spectrum produces the overloads.

PUBLICATION REVIEW’

This report has been reviewed and is approved.

FOR THE COMMANDER:

Chief, Strength and Dynamics Branch Metals and Ceramics Laboratory Directorate of Materials and Processes

in WADD TR 01-177

TABLE OF CONTENTS

I* INTRODUCTION..

1. Probability of Failure.# ^

2. Probability of Survival! «liability.

II. SAFETY ANALYSIS FOR FAILURE UNDER ULTIMATE LOAD .. .. ..

3. logarithmic-Normal Distribution.

^ Extra"e Value Distributions .

6. Multiple-Member Structures..

7. Probability of Failure Under Combined Loading .

8. The Use of Separate Load Factors.

IIL CMSI«T

9* Fatigue Mechanism .

10. stress Interaction and Cumulative Damage Rule .

U. Scatter in Fatigue Ufe Under Ramiom Loading...

IV' ^SId^ME A”D »WMWE

13‘ 3

l^e Distribution of Damage ...

13. Mathmatical Formulation of Risk of Failure ...

16. 'Numerical Example.^

V. BIBLIOGRAPHY.

WADD TR 61-177

PAGE

1

1

2

3

3

5

7

9

12

15

19

19

22

75

27

28

28

29

30

3L

36

XV

LIST OF TABLES

TABLE PAGE

1. Relation between Standard Deviation/’p(^5) of log R (log S) and^Coefficient of Variation yr^vg) ba'sed on Mean IT(S’) or Or/RÍcts/S’) based on Median R(S; of R(s) for Logarithmic Normal Distribution...... ..pô

2. Standard Deviation of log R ± log S as a Function of Cg/S and or/R for Logarithiiiic Normal Distributions of R and S ..38

p. Relation between Probability of Failure ?£ and Central Safety Factor »>q for Logarithmic Normal Distributions of R and S. . . • 39

4. Relation between Parameter ot(ß) and Coefficient of Variation vr(vs) on 'iean R(^) or OpJÎ((as/S) based on Characteristic Values R(s) for Extremal Distribution of R(s).. . 40

5. Relation between Probability of Failure P£ and Central Safety Factor Sq for Extremal Distributions of R and S.41

6. Ratios Tp and Sq for Logarithmic Normal Distributions. ..... 42

7. Ratios rp and Sq for Extremal Distributions.43

8. Improvement of Material Control (in Terms of Decrease of aP/R) as a Function of n to Ensure Constant Probability of Failure p^ = 10"° of a Non-Redundant Structure of n Members (^q = 5.0) . 44

9. Increase of Central Safety Factor Xq as Function of n to Ensure Constant Probability of Failure ?£ = 10"8 of a Non-Redundant Structure of n Members. 49

10. Separate Load Factors ú¿q and as Functions of Probability of Failure... 46

11. Means m^Q, mpg and Standard Deviations C£q; 0rq (in Terras of G*) of Stress' Effect Function.4?

12. Load Spectra for Random Fatigue Tests of AA 7073 Al.48

Ip. Inverse of the Slope, ^ , of log s - log m^ Diagram.48

14. Stress Interaction Factor u>£, Mean m|o and Standard Deviation 0*0 (in Terms of G*) of Modified Stress Effect Function.49

13. Comparison of Standard Deviation c,,q Escimated by Various Methods.3^

NADD TR 6I-I77 v

TABLE

List of Tables (continuée)

PAGE

Load Spectrum for Random Fatigue Test of AA 2024 A1. 52

17. Probability of Failure Pf(N) as a Function of N. 52

18* Survivorship Function associated with Fatigue. 53

19* ÍP FUnCti0n f0r the ^"ation of Fatigue and Ulti-

uJÜ TR bl-177 vi

LIST OF FIGURES

FIGURE

1

2

3

4

5

6

7

a

9

10

11

12

13

14

15

DistributicH Function of V with Logarithmic-Normal Distributions of R and S .

Probability Density of M with Logarithmic-Normal Jistnbutions of R and S .,

ofStRÍband>nsFU'KtÍOn 0i V Distributions

Density of V vdth Extremal DistribuUons

sâtÍ0FaofWen of FaUure pf Central afety Factor \i 0 and "Conventional" Safety Factor v

with Logarithmic-Normal Distributions of R and S .

ífl\tÍ0S b?tWeen Probability of Failure Pf and Central t factof V o and "Conventional" Safety Factor v with Extremal Distributions of R and S .

Decrease o£ »f "Serial Control (in Terns of Decrease of ör/r) as a Function of n to Ensure Con¬ stant Probability of Failure Pf = 10’6 of a lton_ Redundant Structure of n Members (\Q = 5.0)

Increase of Central Safety Factor \Q as Function of n to Insure Constant Probability of Failure Pf = I0-6 of a Non-Redundant Structure of n Members .

Short Column under Combined Three-Dimensional Loading ..

Short Column under Combined Two-Dimensional Loading .

Allowable Increase of Design Load for Horizontal Acceleration due to Earthquake as Secondary Load .

Schematica^Hlustration^ for.

pKmtyÂïe A. .

Representation of Constant Amplitnde Fatigue Test Re¬ sults on AA 7075 Aluminum in fk - N Coordinate ^ste”!.

Schematic Illustration of S-N and S-diq Diagrams.

PAGE

%

63

64

73

74

81

88

89

90

90

90

91

91

92

93

WADD TR £l-r?7 vii

List of Figures (Cont'd)

FIGURE

16 S-mQ Relation for AA 70?5 Aluminum.

17 ^Presentation of Random Fatigue Test Results on AA 7075

18 1¾ - Relation for AA 7075 Aluminum..

19 ^ “ VR‘VS Relation for AA 7075 Aluminum.

20 Extremal, Logarithmic-Normal and Proposed Survivorship Functions for AA 7075 Aluminum under Random Loading.

21 Probability Density, and Risk Function of Extremal Distribution.

22 Distribution Function, Probability Density and Risk i unction of logarithmic-Normal Distribution.. „

23 Distribution Function, Probability Density and Risk Function of Proposed Distribution.

24 Relation between Reduction of Static Strength and -ycle Ratio (W. WeibulllS and J.L. Kepert & A.0.6paynel9)

25 Schematical Representation of Exponential Truncated and Full Load Spectrum....

26 Representation of a Random Fatigue Test Result on AA 2024 Aluminum in ]T^-N Coordinate System .

27 Probability of Failure Pf(ä) due to Extremes of Load Spectrum as a Function of Number N of load Application ...

2! .^ÍÜ. and «« — a™“*

PAGE

94

95

96

97

98

101

102

103

104

105

106

IO?

108

WADD Tfi ¿1-177 viii

LIST OF SYMBOLS

a', a", ã

A

Ai, Aw

b

= ( v/ vn) or (V/ Vqf : appears in the approximated form of Fv [y ) associatea with extremal distribution of R and S

= constants

- cross sectional area of circular column

= initial and momentary resisting area of specimen

= (£/ß or ß'(£ : appears in the approximated form of Fv ( V ) associated with extremal distribution of R and S

b',b",c,c»,Cj,d = constants

D

fv(x)

= R-S difference between resistance of structure and applied load

= probability density function of any quantity X

f(q)(f(q)) = difference_between design load Sq (specified resistance Rj and. mean S(R) of the distribution of S(R) in terms of standard deviation ¢^3( (^R).

f(Si),f*(Si) = stress effect functions without and with interaction

fS]/3) = truncated distribution of load intensity

^ (S)

f(W,Ô,0)

Fx(x)

Fx(x)

g(A)

g(W*,0,0)

G( A)

G( A*)=G*

Ge(A)

h, h'

k', k

= distribution of load intensity representing the extremal part of load spectrum

= joint probability density function of W, Q and 0

= probability distribution function of any quantity X

= 1-Fx(x)

= history function

= joint probability density function of W*, $ and 0

= /0*d4/g(A)

= critical value of G(A)

function of ¿3 (analogous to G(A)) associated vãth extremal distribution of fatigue life

= parameters of exponential distributions of load intensity

= ratios_of design load Sq and specified resistance Rr) to mean of the distribution of dead load Sd

TR 61-r lx

í Mn).L'(n)

ÏÏ5T

pC[f(t)]

1¾. mj

'"iO*

raÍ0‘ **0

List of Symbols (continued)

= height of circular column

= survivorship functions

*•*^«*1 vcixue 01 it " vassociated with

ine Nk among a set of observations

= the Laplace transforr of f(t)

k-th value of fatigue arranged in ascending order

= means of stress effect function without

= mi/G* and with interaction

^i.raax

"R

in^thp0^ ^ associated with maximum stress level SL in the spectrum evex ^fmax

quantity such that lfiR gives mean of the distribution of GU) associated with random fatigue

= m

N

N*

- qjUty analogous to ^ appearing in the distribution of

= number of load applications

= reduced operational life tions in terms of number of load applica-

N'

% -

%• 4

“j

P

Pi

= minimum fatigue life

occurrence ofttonrst fill, T“ the i"54“1 ^ ' structure falIure ln ne,,be” of a redundant

interaction on5t-ant stress amplitude without and with

mean and median of the distribution of fatigue life %

“ and*^ ^ween the j iallures in members of a redundant structure

fatigue Ufe associated with random fatigue

median of the distribution of nr

° PXab“ity °f resista"':e R being smaller than

frequency ratio of cycles of stress ampUtude 3.

specified

in spectrum

; TR 01-177 X

Pk

p(v)

Pf.Pf(N)

q

List of Symbols (continued)

= probability of failure of a member when n-j+1 members still

5trUCtUre 00nSiSti1« of ”

= 8t™a“ety °r fallUre °f k*th "“ber 01 a "»"-redundant

= ïïSÂf intenSlty *

^ (V ) : probability distribution of V

= probability of failure

probability of load S being larger than design load Sq

QjM,Q(cj) = characteristic functions

^p(l1 : ratio of specified resistance to central vain« of the distribution of resistance R. ^

r(N)fr,(N) = risk functions

R - V R,R

-x/ R

Ri' rn

h

^ = Vin

s-i i.rain

- V sf s

= resistance of structure

= mean and median of the distribution of resistance R

- characteristic value of the extremal distribution of resistance

= initial and momentary static resistance of structure

= mean of the distribution of momenta^ static resistance RN

= specified resistance of structure

critical static resistance of structure

= strength"1^ b7 ^1116 "tress by ultimate

stress ratio associated with minimum stress level S- ! V* i»min

= dSstributio^of T10 0f dMi8n l0ad ,!entral 8f the

- applied load acting on structure

mean and median of the distribution of load intensity 3

3 iSufl10 °f the distribution of load

TR bl-177 xL

List of Symbols (continued)

ä = non-statistical applied load

max = desisn load or load intensity dividing load spectrum into operational and extremal part

^min = ^l^imum load intensity in load spectrum

sd*

sd' ¿'¿

3d' S/

0d,q,2i ,(

Si„

Si,min)Si>max

dead and live load

quantities equal to design dead and design live load multiplied by associated load factors

means of the distributions of dead load Sd and live load

design dead and design live load associated with design load

sq

stress intensity S.(of i-th stress level) appearing at k-th in the sequence of load applications

minimum and maximum stress level in load spectrum

ultimate strength

l-ftajo "VíT(j- o : reduced variable which transforms the proposed distri¬

bution of fatigue life N into normal distribution N-'d0 N-N0 13¾ 0r 7¾ : ^uced triable in the distribution of fatigue

R = return numbers

V(VS,VR)

w, W

^d. wd

^'des

w*(e, ¢)

z

a

= coefficients of variation of the distributions of resistance R, of applied load S, of dead load Sd and of live load

= characteristic value of the extremal distribution of fatigue life (associated with constant amplitude and random fatigue)

= total load acting on circular column and its absolute value

= primary load acting on circular column and its absolute value

= dead load of circular column and its absolute value

= live load acting on circular column

= design load for primary load

= critical load in the direction of (0, ¢) in circular column

= section modulus

scale parameter of the extremal distribution of resistance R

WAJD TR 61-177 xia

O^R

fil’ fV

7 9

0

List of Symbols (continued)

= scale parameters of the extremal distributions of fatigue life associated with constant stress level and random fatigue

= load factor for dead load associated with design load S q

= scale parameter of the extremal distribution of load in¬ tensity S

= load factor for live load associated with design load S q

= : ratio of mean of the distribution of live load to mean of the distribution of dead load

00 i r -u x-1, = J e u du : Gamma function

0

= standard deviation of the distribution of the difference between log R and log S

= standard deviation of the distribution of log (n - N0)

= standard deviations of the distributions of log R and log S

= accumulated fatigue damage

= critical fatigue damage

= 1 - A

~ : Parameter of the proposed distribution of fatigue life with reduced variable t1

acceleration due to earthquake and its horizontal and verti¬ cal component

- magnitude of and f * H J V

= design value for horizontal acceleration due to earthquake

= reciprocal of slope of log S - log niQ diagram

= angle (Fig. 9)

= constants

= constant

= k-th moment of a statistical variable

= R/S : ratio of resistance of structure to applied load

= ' conventional" safety factor \S \S A/ -V

= R/S or R/S : central safety factor

kADD TR 61-177 xiii

y', V' r ’

f

a2’a3'%

an’av

ad>a¿

List of Symbols (continued)

= VS, %/S: ratios of initial and momentary static resist- anee to load intensity of entremal part of load spectrum

= reciprocal of slope of log S - log m0 diagram

= radius of cross-section of circular column

standard deviation of the distribution of the difference between resistance and load e

= standard deviations of the distributions of stresses t X, and their sum 2’

= standard deviations of the distributions of f and Y H

standard deviations of the distributions of dead and live

o'., a* i

ai0,a0

ai0'a0

RO

Op %

T = TE or TF

Tr T2, t3

TaO' Ta

= vUhttertti«“5 °f StrCSS °ffeCt £unCti°" ^

= a./G*

= a*/G*

= standard dovlations of the distributions of fatigue lives

i R

“ :ídnapaprUedV^dlT °f tte ÍiSt£lb“£i- o£ «^«ance R

= ^ntity such that tíãR gives standard deviation of the istribution of g(^) associated with random fatigue

= õ/g* il

= standard deviation of the distríhnfí resistance RN distribution of momentary static

= stress at eritical point E or F (Fig. 10) in circular colu™

= a£ bottom secti0" of circular column due to orimary

earthquake ° VertlCal md horiz®“l acceleration caused by

= material strength parameter

= allouable stresses associated with primary load and combined

T* T*

W^A; stress due to dead load in circular column

= critical compressive and tensile stress

= angle (Fig. 9)

VJADD TR 61-177 xiv

List of Symbols (continued)

0^( A) = monotonically decreasing function of A

$ (t) = /t e"u/2 du : normal distribution function

= i- |(t)

f(y = mean of f (tjj)

= stress interaction factors associated with mean nn and standard deviation of stress effect function f(Si).

WADJ TR 61-177 XV

I. INTRODUmnM

the pf°ba^ilistic interpretation of the concept of structural safety and

S «2 S6 ?eterMnfíÍOn °r safe^ factors has btntv" loped Thireo* h!fL t as 1,611 as in seïeral &a-opaan countries 2.

COnCerned ”lth Saiety r6Sp66t 10 a =1^16

1nArtp?ie+inCraSing s®verity of the operational conditions of modern, dynamically comblned ^ the use of structural materials of higher Static"

ulUmlte ÏÏLtsîrn íati8“f r66istf66 has squally Changed the emphasis from to f i?".10 ?e5isn ior faU8"e- 1,16 necessity has therefore arisen to extend the statistical approach for structural safety to the fatigue design of mechanical systems, or rather to develop a statistical procedure o/safetv ^analysis

thSXelt fi0n °f ^11^6 l0ad and fatleue which such systems, íhetheí designed.slruclur6s' :iachlne P^ts or mechanical parts of control systems, must be

The considerable improvement and refinement in the methods of stress-analysis

wav rnTt^b TtenS P0SSlbl6 ^6 lMr6ased of computers is ï no

based on fhe use^r6"6 Safety i" general, is still justified^ ^ ., ° e °r a” 8afet7 factors that enn neither be j tilled by rational argument nor related to a probability of failure It is +hp

™t£aXloÏÏt-Î0 'T1”“’ 0" th6 baSl8 01 r666"‘ work concerned m^n ^ ^ ^Mect WaV ï1Ure 3' the P1-1"61?165 of safety analysis for structures Terilld? operational loads as well as to an occasional excessive

load Fajlpre. Let S denote the statistical population of ad intensities that can be expected to act on a structure, with probability

density fs(S) and probability function FS(S), and R the carrying caoacitv (resisUnce of the population of nominally identical structures,with probability density lR(a) and probability function fR(R). men the probakut^fÄe

Pf = P(R<S) = P(R-S<0) = P(R/S<1) (1.1)

provided b >0. This probability is the stochastic limit of the proportion of tructures out of the population of resistances Fd(R) which will fail uhpn a i a

!61!^ aî ^ f-Vhe P°PUlatl6" r3(S) is tlied a" u Xe ecte at random from the population Fr(R). i>eieci,ea

. Probability function Fd(D) of the difference as the marginal density of the joint distribution of D The probability of failure Pf = FD(0). Hence,

D = (R-S) is evaluated and R or of D and S.

Manuscript released by authors .March 1961 for publication Technical Report.

as a WADD

WADD TR 61-177 1

(1.2) 'O Pf = % FR(S)fs(S)dS = Ç F,(R)f

(S. 8 > 0), a„d FS(R) , ! . K3(r).

evaluated as the •’lr(iV i °î V = R/s 1S density of the joint distribution of V and ' S 4~:

Fv (v) = ^FR(VS)fs(s)dS (1.3)

and therefore

Pf = Fv (1) = Z* FR(S)fs(s)dS (1.4)

since by definition1SFß(0)O=SF^ (í)"tlCal th® Second ^ of Eq. (1.2),

not- a measure of^the safetv^f aR!+ 'P16 probability of failure P

lo ids from the population FS(S) duri^its^^^01^10 a random sequence of f xs ns probability of sun^i L(Nf ^nler ^ 7 Such a raeas^e the sequence and is given by ^ ^ load plications constituting

N KN) = (1 - Pf) (2

ditions of failure^de^rsingle applicSio11^^1!8 limitation dei>ines con- íoSÍ7nr 0f the structnre is unaffected bv^nv load such that the load that causes failure. d by ^ load Preceding the "ultimate"'

Pf « 1, Eq. (2.1) can be approximated by

which is failures; condition

1(N) = exp(-NPf) or

thn/p1J'"known survivorship or U/Pf) = Tr represents the »

InL(N) = - NPf (2.2)

"reliability" return number"

” function for chance of failures based on the

load-applications («fatigó«») or^th^e duratiön"»?i|nCrefinS "'»ber of

o* ? : ii'frof ( wi rtv ior'rg( cr“p-rupti're"^ A f definlte trend of chan-e in servir. . 3 s) Hy 06 » function

pected.) Therefore Pf becomes action 0f '1 ase must «- is replaced by i unction of J (or of time) and Eq. (2 1)

IW = £ f i.Pf(N)] (2.3)

WADD TR 61-177 2

or, since Pf(n) « 1

UN) = exp f - ^ Pf(n)] (2.4)

through its effect^n ^Tv) anÍ ^w! 0f FR(R) ^ N

.xs¿: yrÄs' isâ»,“». ’ “* " »“■«« » iSSiFSSÎi,' ^

The relation between

or

UîJ) and Pf(N) is obtained from Eq. (2.4) :

- dlnL(N)/dN = Pf(N)

In general,

Un) = exp f - /N pf(n)dn] (2.6)

(2.7) r(N) = - dlnL(N)/dN

is called risk function (or intensity function 5).

11 • SAFETY ANALYSTS FOP

by asXdSÄIe“ott(Sr 1°* ar. developed and by introducing the t«o fo^s of tte futons n "r aS of “• been found to provide the best representation ôí o, !S i Í .FJ.<S) that have of relevant material properties and of ranrinm i °P°erve^ statistical distributions distribution and the asymptotic distributions of SSeme

3- Mhrtic-;^ rib,„ions. Introducid the following ^eUons

Fr(r) = {(iSO/I, On (3.1)

and

where

FS(S) = § (lQ£ S/S)

= ~ÿW exp ("u2/2)du

(3.2)

(3.3)

u and 5 denote medians of 0 and S and £R = (f(iog R) ^

WADD TR 61-177 3

cT: = cr( log S) arc the s it is obvious that the nor.iaj. ^sincc the disir -°3 (d/S) and standard

tanuard deviations of log R and log S respectively, distribution of log ^ = log (u/s) = log R - log s is imitions of log R and of log S are normal) with mean deviation d = Hence,

= ^[(log^/^)/^] (3.M '■•Here ^ _ R/S is the central" value of the safety factor, and therefore

P£ = F„(l) = ^[-(log^J/c;]. (3.5)

>c'-sion if eStablÍSí ^ relati°n2 ^etween the measures of dis- of variation v <Ti{ and ^ and the coefficients nL Z * ' = ~ 7' - VS = as(s /ïïwhich are usually known for actual ooscrvations where R and ? denote the means of R and S.

It can be shown that

\ = (°^)2 - 1, V® = e>.T . 1 (5.6)

where a = In 10 = 2Ò The relations

v.P (a./.r = V, exp (a/)", (o^/sf = v2 exp

J.?)

permit the conversion of the coefficients of variation into the ratios (a /r) aaa based on the aedians. ' R' '

. ori-icr t0 illustrate the safety analysis, i-afos fOn/if') _ n nn n in •md 0.15 « well as (as/S) . 0.10 0.20 and 0.» have’ be n'lsfuied íabk 1

L ^ - ::S0ClatCd valucs o£ 4½) & vd(vs), Table 2 rkuUta» lues g fo. various combinations of (ag/R) and (ag/S).

- i / rq* thc iunctl°ns ly(v') for various combinations of (aD/R) am (as/î l ave »eon evaluated „it!, .¾ as paranetcr and presented in Fi°s l(a) to l(i) „here r >/) = on logarittaie scale have been plotted arhnst

V. The probability density of lo8l; for j,/Y = oc/q' - n in ÏC f a^ainst illustrating the fact that the conventional assumption that some Estant11’* " value of the safety factor can be associated with a structure is meaningless- any value of 0< y>< œ is possible, although valucs in the central range ar¿ much more likely. The range 0 < ✓ < 1 defines failure, the ranve ^ ? l°sur- vivai Therefore the probability of failure Pf = fv(1) can be read ¡irecUy

f0r 3 Specific rati0 of at the intersection ^ ~ Fl8- i). Conversely, for specified values of (aR/R), (on/S) and Pf, the required ratio ^ ("central" safety factor) is obtained! S j

’.:ajd tr 0I-177 k

course to gr ¿S ' repre s en tatto ^ b® .used„direct1^ for this purpose without re- with various ratios ^ /f * . Resents vaiues associated by using Eq. (3.5). ^ l/K/ ( ^ J. for duferent leveïs of Pf obtained

For a nonstatistical load S = ■;* ana <J^-o rn ¿ ailJ Tq. (3.5) reduces to

r f i [-(logV0)/f,j (3.8)

where Vo = R/s*.

(or «elbulosilllî^t vâu«d6Cf"j: ‘“r1 asymptotic (or Frechet, distrib^UtrJest^iues ^

and

Fr(R) = 1 - exp [ -(¿r] R

Fg(S) = exe [ - ^ J

(4.1)

(4.2)

dtwbutio“f Je 7 z"cTctrsr^s (ciose to "°des) °f the the standard deviations (5(log r) arri 6fUnCti° îE of is obtained fron Eq. (1.3) in the fom S)' ^distribution Fv;V)

^(V)=

»her. t = (S/S)-fi and v0 = R/3- or, alternaUveiy,

By the substitution u = ex,(-t), (¡,.3) is transió,»cd into

h (V ) = I -/e* p [- (i )* (_ („ u p ■£ ] du

Smpted atoflmt;rVenlent- f0r nlMeriCal eralUati°" Proïided ^

Tne probabiUty of failure is therefore

(4.3)

(4.4)

(4.3)

is not too small

WADD TR 61-177 5

ff* Fy (I) s I -/exp f-t - P ]dt

I-

or,in terms of u,

-/ e*p .

(i relations between the

(4.6)

(4.7)

efficients of varlaïionT vR = K ^ ^ ^ the and 0*3/3 are obtained i/tho ?oUoSg fl™. 37 the raUoe <r R/S r/S

CO*

and

as well as

and

v*[rú+£>-roi-¿)\l/r(í+±)1

vg - rro-^-ro-^rG-^)1

<Vk)1= r(i+h-rb+-k)x

(tsi§)x, rO-h-ro-j-Ÿ

(4.10)

(4.11)

Using the above equations the values of Table 4 (a) and ÍM „ u u; and (b) have been computed.

By numerical evaluation of En (U O <> a* various combinations of (<hR/R)\d ( V% T f Fv(v)eP(v*) ior v0 = R/s as parameter and are presented^!nFi co^síructed with

evaluation use has been of the app^^-o^t^dStand^^V

Fv (w) - ar(i -b) (4.12)

nr1^/ the SUbStÍtutÍOn b= *//* »>• provided a « 1, The p

WADD TR 61-I77

n u0r,.//K (b<1) and a prooability density of V for V

6

CrR/R = ^“3/S = 0.1 has been illustrated in Fig. 4.

The probability of failure P^. = Fj/ (1) can again be read directly from the diagrams P(y) at y = 1 for any specified "central" safety factor y

values of associated vdth various ratios of ((Tr/r) and V for different levels of Pf, obtained from Figs. 3(«0 to 3(iJ.

In the particular case of a non-statistical maximum load intensity S = S* and therefore ^3 = 0 and <T3/3 =0^ Lhe probability of failure is obtained directly from the reduced Eq. (4.5) introducing v'*!:

Fv(vV I - <=*p ] (4.13)

where y = r/s* and = T/S*, and

P-f = ' - e*p[-V0“ ] (4.14)

vdiich is easily evaluated.

Comparison of the values of ^in Tables 3 and 5 at the same levels of P_ associated with the same ratios of (O'^/r), ( O'5/5) and (<rR/f), ( <rs/s) r respectively, shows the differences resulting from the assumptions of logarithmic- normal and extremal distribution of the variables R and S: for extremal distri¬ butions much higher "central" safety factors are required to ensure the same prob- abaility of failure as for logarithmic-normal distributions. It should, however, be considered that the load-oopulations considered are not the same. The extremal distributions represent partial populations obtained by selecting the largest values in samples of the whole population (gusts exceeding a certain intensity, flood levels, etc.), while logarithmic-nomal populations usually represent the entity of loads 3 > 0. Thus the number of applications of extremal loads is much smaller than the total number N, so that in order to ensure a specific value of L(N)a much higher value of can be accepted.

With respect to the use of extremal distributions for the representation of resistance properties it has been shown 8 that superior production-control is likely to result in logarithmic-normal distributions, while poor control leads to extremal distributions.

5. Safety ^actors for "Maximum Load" and "Minimum Strength11. In conventional design it is usually assumed that the safety factor can be based on a "maximum load" and a "minimum resistance". However, with the exception of non-statistical loads, such as the fluid or bulk-pressure in storage containers or floor-loads in warehouses, no absolute maximum can be soecified; similarly, no absolute min¬ imum of the resistance values can be known, only values representing the smallest observation in samples of finite size. Thus a "minimum" resistance will always be associated with a finite probability p of not being attaine^and a

maximum load with a finite probability q of being exceeded, however small these probabilities ar^ selected. Thus = iL = rJf (for logarithmic- normal distributions) or rQR (for extremal distributions) and

WADD TR 61-177 7

ä&ibÖ.3

)ns by the equation ibe ratio r is relatpri_+^ °r Sq ¿ ^or extremal

ir.al distributions bv t.ho oR„*+-!^_ ln case of logarithmic-

p° iLflo, '-[>VsR]

ini in the case of extremal distribution

(5.1)

S of smallest values by the equation

■similarly the ratio

p * 1 - ex p (- pp) #

sq is related to q by

n ‘ ? ui=3

(5.2)

where $ - 1 _ a .., * m the case of logarithmic-normal distributions.

i / ^ - I - exp (-Sq )

m the case of extremal distributions

(5.3)

and by

(5.Ü)

Of largest values.

The "conventional" safety factor v is „ow defined by

(h.5)

■ind is thus related to the "central" f . w

ina V For a non-etatistical maximum load ;« ^ the selectai Mtios ^

5- ep/S*=v0vp (5.h)

approximatiorTthan^p^0, l5 idth^’iéf"6 ^ U speoified » lle^1: ,, “1«.^? specimens

It is, with better

the probability* oft/alues ^ «*»^^"*9°^ analysis of actual acceptance tests of str !" Vaiue of the sample is 0.1

percent of the observed yteld-stress values’fa^ bet^tîe^spSiS^r )°

currenee muni,er of (l/q) of loads exceed^ selected as small as desired

jq depends on the length of the is be determined. :-.1 th a re- Oq, the value of q can be

X L

if the series of observations is long^no'ughï

For o = 0.1 and q = o.l o 01 n nm ja have been computed using Eos h ¡i f; ?;°?? ^ 0.0001 , the ratios r and s P°™l distributions and in Tables 7 c-

WADD TR ÓI-I77

peif-n0 ,,¾¾ thf f of Tablês 3 (logarithme-no mal distribotion, urp r f5 5 (extremal dlstrlhutions* d = q = 1 - l/f) the relations (¾ 5Î an 0?5ï6)10U;00'1,bin1ati0"5 0f (B°') '«le obtained bj the use of

T and C5.dJ. These relations are presented in Figs. 5(a) to 5(/) for

bE diStrib“tiMS -d 1" n«.. 6(a) toV) for e^rLí oistri-

P îf’thl pmbaMn?;3,96"?“ the Sel6CU°n °f safety factors mated +u b b level q associated with the »maximum» load can be esti- associated ^ ^ * ?*“ ^ lnterPreti"S «.e probability of faaSre associated with current safety factors.

Consid6;ritrÍÍs(mc'rb<:r <a> Structures „ithout redunda,Ky. considering a structure consisting of n members each of which is -ubieot to the same load (total load divided by n) and assuming thaf the stÆ fails

, y one °y lts raemt)ers fails, the probability of failure Pf of the strurturo under a single load application is Pf = 1 - , # (l . D ) = ^ „ ,StrUcture is the probability of failure of the k-th member. k k=l ’k h 61 e Pk

and iffthp°babllliieS 01 failure pk “ ?f/n are assumed equal for all members e sa V Lh~a Pr0bfb(Ll^ 0f failure Pf 15 to be retained, it is nec¬ essary to reduce the probabixity of failure from Pf to Pf/n As-jninr In anthmic-normal distributions for 5 and, S, this (eductiof, requîms

1) decrease of (r ;/ii beeping „„ = r/s and^ o-./J constailt; 2) increase of 0 --,- „ ¿I so Wl

Keeping (r3/o and (T,/R constant; V () 1 1 V' .A. j

3) combination of (1) and (2)

Operations (1,, and (2) ire discussed us U" SnS, (5 1 ) tnj ( c r.., +- ,.,.

Of statistical und non-sUtisUe.l load

pK-# t-(ng v0)/s] ;C )

PK = $ P Use, O l/éj.] Í0.2)

tu* oïtufe CJ7XâJZhr “ÍV-Tír thV?"Cified pro1'- given values ofer./S1 and V ,J^^ber md Pf of the whole structure for

10- 0’ imples have . een computed for the issu;aptio.,s (i ~ L¿". ’ , v0 = and ff-j/S - 0, 0.10, 0.20 anu 0. '0. Values 0 thus^obtained a-e listed in Table ; Fig. ? eresents the relaUond/VcLe.-,-,

n. =3 and n, illustrating the necessity of reducing <rR/v

ad (2); Values of

as

(Tn/R

increase;

0 ability of failure oder given for Pf = 10- <tr/r The results are shown in Table 9 and in F Oi increasing ^ as n increase

re v-omputed whicn would guarantee the specified prob- ¢-¾ R and J3/5. Examples have been com a' '

and " 0 ,05, o.i , o.r ^5/^ ~ c, 0.10, 0,2 :. 0. - which illustrates the necessitv

F ogs. 7 and 8 illustr ite the 1 U VC: effects of improved control or of increased safety factors o t r-'V”' „'r

redundant h Itinle- ,ember structures to a wdníd -Äut* (ft

V’ADD IR 61-1 7

lhe ^ivoPSÄof/o7iVal 0f 11 ind^<*nt load application^ ‘ 5tn,oture °í » manbers abject to

Lfn) - Uxp(-Mpk)] =e)íp(.Kjp^

Pf = n pk. 31 Mlth that of a Slnele m™b«r structure of probability of failure

ufpröbTi-'t^1'"1“1"6'"“^^ ^“faïtofof r(7í fr“” the faU“re °P the

ConSt j ^ uniikejy, though not impossible.

the total lo id is always Equally distrih^^^^ Under the ass™otions that (a) resistances of all ^bers’beLX to ine ^ T"16 the e)dsti"e »sr.bersf (b‘ ^

f ns bL“J- íí6 int^al w^on i/oT^i5^031/0^1311“' « ïf~s ¿* .Äxr z a -íÂstssawsr - that such a meníiPr vm o ’. nen vn-j+i; members still exist , , , Pj « 1) e^S;pr;m — -¾ load applications1“

exp [-(n-7)^'ï n‘J,fl ■iecibers will survive ioaj ,. densiti/ / i -; ^rom which the probabilitv r + J ^°ad fPP-^c^itions is density function ¿q. (6.4) of Nj arç obtained.V ^ 0 (0-3) and

F"i ^ * I - txp [-(n -j + ÙNj p. PjJ

^ = exp Nj Pj j

(6.3i)

(6.4)

The corresponding characteristi c functions are

QjW= •( exp^NjHN.^dN-, = O J J J - - ^

(6.^)

mo > -¿n-j+O p.] t

Then by the virtue of law of

N = £ \ is

WADD TR 61-177

convolution the characteristi c function of

10

(6.6) qP- TV i

K*> I - iu>

(n-k-*-l)p

"Isliven ^ F0UrÍer inVerSi0n f0rnula' the P^bility density function of

f H ^ * ttr / Gi (ui) .xp (-UoN^di -CO

XT r- i

3 to

I I uo I - -Üâi- I _

nP. Co-i)Px I - ±±

Pn

exp¿ítoN)íiN

where

(6.7)

which is easily evaluated by the method of partial fractions,

fNM = Of. C evP t-r,R Nj + (n- ,■)exp [- (n.^p^N]

+Pn^^P^Pn^ (6.8)

or the survivorship function is

L(uV 0, exp L-np N]+ Cjexpl-in-i^vd +...+ Cnexp I-pnN] (6.9)

CJ = I - j y Q pj ( _ Cn-¿ + i) pj

^P, (n-i)pi , _ (r>-4 + \)Pj.

ín-j -»-i) pj_)

) _ (H-'ù+ Opi , _ (ri-^-t-ùp.

whichCT&trre^SthS (SaîfloaadSr00tT thre6 ia bated with <rR/ÏU 0.05 (SB , 0.02166) and ^/1-^ resoectively v0 = B/S is assumed to be 1.779 in order thatJ P1 = 10-7 P2 - 6.03 1° and P3 = 1 - 1.26 x 10-6 . Slnce furt)ler C0BpPJta“n ;hous

WADD TR 61-177 11

that c] = (1 + 2.?9 X IO"6), c? = - 2.32 X 10“c and = 4.00 X 10’

L (n) = (¡ + 2.1^)(10 )eXp[-3xio Nil - 2 y lo"^^ Í -i

T ^yzxio exp [-, 2o¿y to Nj]

+ 4.00X l^^ex p [-6.-,. 2.4 X /o’Sn] ^-1°)

There

stup function L(n) = e-3 x Mr’"» assocLt eqUatlo,i and »• survivor- dundancy. Important, howvor, i. the ?âct n *”Tïh ^ StrUCtm «f no re- suryives »Uh a reduced prooabilitv of f , lias not failed but 1,1 the ^ve" example, the survivorship r^'r*1 eVer after om me"b'!r fails: measured from the :nstant ,,i 'hlp ^u'lCtion in terms of the time f t) 4(11.) V e-l^irx^ifj,“- 0- Pf the members fails is

nost immediate collapse of the entire Ín cT °f.the.sec^ «ember causes al- •L^‘ F^°m the engineering point of viev, ' theref 10 VleW °f the large value of urcraft design should be based on the éonÍfa ^f6’ 3nd f°r structni'es such is

^ 31 ure of one member should have a probabilit^f00 that the structui,e after efficiently close to unity for th* p °baballt^ of survival ^(w*) = e-l^OóxlO'V ttm» U, «-¡«d Jp«,U«a Um If (flight time

Tr urcted ^ the point »he4 L00fUariSl d50 71 the StrUCturi; «•»

considering its ejected SLl iffi* of survival of the structure after fauûre of u bS,m>tched ¡V ‘he probability reduced operational life" the W^+ \ f,lts first member, considerinr as

°oint of repair, the redundancy of the =.+66 necessary to reach a

T$\kST* a rati° °f oapSeîaUtIy aS ~ally in the qa 10 ’ the P^bability of failure of th/ ,atl°nal bafe of the order in the same proportion by the failure „r ? r f th structnre may be increased ducing its "operational" safety (orinciil^ V redUndant raember without re- may oe worth noting that by the fail-safe" desi-n). FinaUy it

increases. ' = A h ^

—L, -Wit, uy Wie Virtue Of the central n ■ .

= & \ approaches a«ically a no™¿ ^i s^ n

rrnbililil" ^^-bure Under Combi peri -,.

The manner in which thQ « aithough u particular ^ -^atablished is puite general,

‘Pne principal load ward, and of dead load due to earthquake is the acceleration due to

The total loid is magnitude W, 0f T~ •AdD TR ÓI-I77

«P is the sum of live load acting at the mass center whi +h 6 t°P acting doi

vd acting also at the mass c¡ntír -h^ Sec-2£daiT boad earthquake and w, is +Uo .‘.er’ 'here * denotes

'vd as the weight of the colu- .. W = aod

tt * wd + 5 d . the magnitude and

Since it direction

is^assumed that the °r 5 are statistical

12

and1^05^1^0" Sft^t7) T?heledomSnS 0 Tzr and 0 < ^ where W is the absolute value of vt (Seî Fig. 9)/ ° = 9 < Zr

(a fl(ïf ^ Can be assui"ed that the critical load W*(e, ¢) in the direction (, ¢) is ^nown, so that any load larger than W* oroduces failure of the structure, the probability of failure pf is ‘ exaure oi the

. 2.ir

^f=/ d©^ àtf> j {(W,©, Ç^lclW .

If ’/(9, /) is also a statistical variable,

(7.1)

H d©/ / çOdvr/ f (w, ©^)dw ° ° ° J w*(e50)

(7.2)

where g(w+, 9, ¢) is the probability density of W*. 9 and ¢.

concent o^aïlo^bf EqS‘ (7*2) a the°^tical justification of the concept of allowable increase of design stress for combined loads can be riven Design specification of bridges as well as buildings, for instan e pemits increased design stresses when secondary loids such as wind and loads due to

SSÄT S Wlt“r WU- with the principa! loa!, /uch as Uv7 load

ani simplicity, it is assumed that W*(e, ¢) is independent of </>

respect L he z al"^ the occuprence of f - Metric with PnVP +ï / the pro5lem ls reduced a two-dimensio ial one (see Mg. 10) with replacement of f(w, e,t)d<?> in Eq. (?.l) by f(W, 9)d^/(2nr,':

^=/ d© f \-(yJ ° w*(e)

(7 o)

•3) is hardiy usable because ^ y for horizontal

o.n spite of its apparently simple form, Eq. (-7. 3) is h ird it is difficult to evaluate f(W, 9). Writing and f 7

S tt. sues;r°uet °f filare L'»r.puted hy consider- that V ind V p+ iv d, '“l ^V'^d separately under the is sumptions nat y and y ■ are mutually independent statistical variables with nonml

joint probability oensity Eq. (7.b) and that WD = Iw ] = Iw + VTl ] a- m exponent!.A probability density Eq. (7.5), MhereP r p= ± |'d

j L+- — I ^ yl ; positive si,'ns are taken when tie direction. )f '■e -.,, j 5. are identical with the positive directions of x and z axis , ¿spectively).

'H’^V n v 27r<rH<rv

V/ADD TR 61-17

(7J0

13

■fwp ^ = ^ expT-^CWp-W^J (w > wd) (7.5)

section and its probability density is ^ distributed on the bottom

fT( (T¡) = Alo- exp [-AuCt, - Td)] (7#6)

Vn er e \fi / a

x2 due to f ^ is 0f the co1”". a. stress

T2= ^V^cl ^ e (7.7)

T =h-^«abili^^ 2ero mean and s

over the bottom section. Deno?Lg by ^ ! h ^ due to is linear E and F respectively (Fig. lo)f 7 ^ d T3F stresses at points

T3E--V^HW^/(zZ)^MTj (7.8)

wh^re Z is the section modulus and 'rt - o Û ~ / t

™ radius of the circular coluinn. The statistical^' \ •¡^+and/> beinS height ^3P are both normal with zero . V . , 1 dlstribution/ of ? , La

He Ce’ ^ total stresseÄr f ^fre^^ ^ = ^ Vf. ^

Ts = Ti ♦ V t3e

tf = ti * T, . r3F = T, . T, . T 3E

^ aretdeutt

betting r represent T E and ^ (

^W-Ahevpr-AuCT-r^+AV^/z]$[(r.Td.AUa)/<r 3 di Act

(7.9)

F^Mfo-T^l-^p Ak„.tVir , u i3 Z3J

,_ (7.10) »here t2} = Jt¡ + .

finally, the probability of failure of r> i anuí e of the column is approximately

WADD TR óI-I?? 14

pf = 2[1-ft(t*)+f(t|)] (7.11)

WafrVi and are the comPressive and tensile strength of the material. °

AH parameters needed for computation of the probability of failure are functions of the distributions of loads and the cross-sectional area. In order to ensure the same probability of failure for a given loading condition.

e cross-sectional area must be independent of whether the design is based on the primary load only or whether secondary loads are also considered.

When the design of the column is based only on the primary load it with esign load Wd s and the allowable (compressive) stress Ta0, the area PA is

ously A = Wdes/Ta0. If the horizontal component 1¾ is taken as the de-

~UY£ the SGCondary load> and if the sane area is used the maximum •jtiTGSS Trj IS

w T =

des A

2/I W J , _of r

.3/2 Hdcs (7.12)

rl* indicates that the combined loading requires Ta as allowable stress

(7!l2)ebJc^esr 11 ty °f failure* With To0 = Udes/A and Td = Ud/A, Eq.

}@ ^11, UCS (7.13)

Since Wd and Ta0 are specified quantities, a straight-line relationship is obtained between Ta and which assures a constant probability of failure. The relationship is shown schematically in Fig. 11. This is the theoretical oasis on which the increase of the allowable stress is justified. Although a cinple structure and a simple load combination are used for illustration.°the concept underlying the above discussion is quite general, ''

?-h-?--Usc 0.f.-b.cParate Load Factors. In some recent specifications^ the use of separate load factors is recommended instead of a single safety factor to be applied to the total load. Thus, if Sd q and SfiQ are intro¬ duced as design dead and live loads it is recommendeà^o uselüparace factors

OCy associated with 5d;q and>5q with SJ>q for the evaluation of the total de¬ sign load to be compared with R , the design carrying capacity, instead of using ^(Sd>q + Sjt>q).

Clearly, separate load factors cannot be related to the concept of prob- ability of failure unless they can be related top-. Thus the problem that arises is the following: what are the representative values of two statisti¬ cal variables x and y with probability densities fx(x) and fv(y) respectively when only z = x + y is given? It is obvious that there is an infinite number 01 combinations of x and y which add up to z. Out of those there arc, however the most probable values x* and y* which might be regarded as representative. ’

wADD TR 6I-I77 15

-ÄÄär

x( >ry(y> - txU)ty(i-x) . tx(.z-y)ty(y) '

^ the roots „f the foUoidng K)Mtions

<i[fx(xHv -o, drf/x-^X (j,)]/, =

(8.1)

which are also (8.2)

respectively equivalent to

i*) _ fy C-Z.-%)

fy (t-Tc)

U--Ù

(y) Ix U-y)

(8.3)

(8.h)

the roots rsdSPe0ta^lysln ^8, «ofaíd^")/ Cthera repla,!ed >>y Sd Evidently, ^ ^ vtioh are considered

R S8 = Sd,q + sXq • By setting z , », r00ts . . ^

^ $d “ ^ ^ obtained euch that

= Sd ^ 5/ *.

(8.5)

^ olarifyqthelnprotieíSthesSe°nu^e ''elated by Eq. (5.5)

are schematioally iílUstra?¡d 1" ^

Sd‘ -d V * in terms or 3d ^ s , d,q and Sj>q ln E(Jt (8>6);

^ = Vd.q 4 /0qSj ( (8.7)

When both and

and standard deviations distributed with means s .- are defined by d and ^ , and y. " Sd and Sj

y J d ’ ^ ' 7 . k» and k

y=v/5 s =k.f -t11' 1 ksd) (M)

«add M06l-i)7ty densit;>r Punctions or sd and Sy are 16 ^

^5ci) =VZr yJ , e*P /(Zvd5d>3

axp ^-^//(z.y^i.

(8.9)

From Eqs._(8.3)t (8.4) and (8.9). Sd and 3j respectively. 1

,q and are obtained in terms of

vjCk'-yHiV _

YÎ+ïV d JL

v]4rav^

(8.10)

(8.11)

oxnce o+ u V , -C ~ Iiwixy aistnouted, 5 _ 3 + s dx stributgdw^th Mean S = 5 . s, art .taidard daíutli

0 s y<rd ♦ Î the design value 3 > q

sn = S ♦ f(q) <J\

«i"?,.o^rie from a tabie °f the error disw-

k' = Sq^d = 1 ♦ ^ 4f(q) M ♦ T2 V 2 (3.12)

aCuo":SVSleVV-d ; 0,0Lar,d ^ =K0:20 for i' two r Sdge. If Tis assuraed°to^be 0.0^¾^ ^tlÍ ^

oVe3. 5ã°SÍned fr0m (8-10) “d Í8-ll) '“g' k-

^,0.01=1.01¾ “"à S t.0.01 * 1M'SJ for r O.0 and

Sd.0.01 = 1-05 Sd and 3^,0.01 = 1.^3/ for r =0.50.

in ^s^tsSnS’sfu) rCSU^ °btalned by rePlaCing k' k

WADD TR 61-177 17

s'lL*rY)+Ÿ ,¾ . 3- *■

Vd V,

v>t

và-hOc-i)yy)

vi

(8.13)

(3.14)

substituting S * anH <3 *

Sd„ and .ith thi aid'oTiS; in Un"S oi

e = v¿. f Vd4-(fc-i)rv|

Vd(t'-y)+yY ^ 7^(^-0^ si,n

Hence by comparison with Eq. (8.7) ,

^c*-y)+ï\r

r \j7*^>y¡+\j-' p<\~-

Therefore, with k* = 5.403 [tf- 3 q)

X _ ^003-5^-34+ O, 3(.

va +-CK-Oy v¿ ^+(<'-0K2 •

0.01 0.344

and with k< = 1.761 (/= 0>5) ?

of X g ooxsôç-ojïVi.^, 0.01

(3.13)

(3.16)

â = ft tx(x.-0+ 0.007^ 'O.oi 0,531 (3.17)

o. ÔI3Z Aoor—^-'^00^^ (g.18) 0.0(77 *v ’

denítl^V^TosS^a^ ass«^ 7™1 and standard

^d^™S;1) ^ = °-W " • ^ace Lin 7 and “s^V^ d^;tri.

pf = P(R“S < 0) = §[-(R-S)/ö-]

“here ^ = fil* o-d2 * .

FOr r = 3-0’ ^ vd = 0*°5. and ^=0.20, '

«• = Sd and for (-,0.5, and

(8.19)

0-20, S = 4.0 Sd, R = i.0?k

vd = °*05 and = 0.2¾

WADD TR 61-177 18

s = 1.5 S. R = 1.07k Sd and S = Jo.OOZQQk2 4 O.0ÎF5 h

(R TO?103’ assuraing k, an nl 311(1 fl’ora ând P«. anH KM „ lo «ï_ , — *f 1 l'ora nq. ^0.19;. Tables^Ja)^0?^ 1î1^d+fr°m Eq* or

Pf. and Plg. 13 shows the factors^0 ni ari^ a°-0;^ ..^0.01 In this way the arbitrariness of th* ™ ,^0.01 as functions of

can be eliminated and the factors associat^wUh^ S®parate load factors ure. However, the analysis for any but norl? ? Probabilities of fail- and carrying capacity is nrohibitiíe It inoearf^°11 functions of loads purpose is served by the use of septate loS ^ ^^f® th3t n° P^otical °f safety. separate load factors instead of a single factor

111. RANDOM rmn.

crite^on0^^6^8^^ thattbe Probability

in each load application arïAhat therefore the ^ t°m dama£e is Produceb the course of repeated load applications will fin oi' Sucb damage in probability of survival will decease mtf ^ failure‘ ^us the the form of the survivorship or "reliability" funM^8 nufber of applications; of damage accumulation as "fatigue" differs then ?" characterizing the process ability" function LW = .,(.¾

J(s) the effect of the’^MenU^tLss inteSitv'T^Tb da"aee' by

ia .Sde.“T ed da”aee (hiStory) - rA/iv,e(tAb[a“Ct

dA/dN = f(S)g(A). (9.1)

Sid = 1. ..:;;rvs II “Zf ^ttade v of stress iata„SitieS probabilities Pi of t Jr 0c0“rZe! frequen<:lM Ominad by the a prion

According to Eq. (9.1) for the sequence of N loads

/q dA/g(A) = G(A) = r'f(S)dN = f( 0 k=1 (9.2)

ofei*M ind?-cates the stress intensity S. which is thp ir • of N load applications. Introducing a critical i® k"th in the sequenci damage at which fatigue failure occur! th! Sf1 A of the accumulated minimum value N which, in Eq. (9.2),^^1¾ =* G\ ^ b® defin®d as the

statisticai^ariabirfoT^i^hV f^Ction* G(A) b^omes a for a given value of G(A). Be virtue of the dÍStrÍbUted

iward normals far 6 n!l_al lljnit. theorem the distri-

Si.

bution of G(A)

WADD TR 6I-I77

will tend toward normality. Under constat stre^ density

19

this asymptotic normal distribute u

co^iderinj^he mean and”Sndíd deriation^f' f(ff10" OU) is glveSV 6 iailUre Criteri»"' ^ ProbatiStXli'fi^Wo«.

ffi)] = ï[(1.miû)/ciofil)]

^th the following expressions for the median M % 4 Ni' mean “i and variance

(9.3)

% = (9.4,

i “i f1 + |( <5f ,/"i0)] (9.5)

< = »i r^o-Kv2] (9.6,

where ni* * — m /rj h * fatigue tests. ' $ $ is ^ ^3—?from the résulta of

The survivorship function ^

LW= ^^4 - ¢[(^,/(^0,]

o> --ngedt LUdin0'5:^0^^ .the fati^e life

(9.7)

Introducing the variable

ascending order, since the ^ted vlL U

L(\) = 1 - k/(m+1) (9.8)

t= (o’-lis.Vff^. (l.lfcio)/(r €

linear relatió. is are obtained

‘ fî“G7ri‘(V'i,K =

between t f? and N> If

(9.9)

10' «io'W

‘k corresPonda to the observation ,1

(9.10)

,vk so that

WAOD TR ÓI-I77 20

= 1 - k/(m+l) , (9.11)

Points f kÿl K ) bt +n®d from a table of error functions. The set of svstä M ™Su baureProducible by a straight line in the coordinate to En /n o\ ^ , lf the hypotheses concerning the fatigue mechanism which leads

«sSís'o9; L™/” ^ïf!rably Vaííd- the evaluation of lift thP 7 L5 alumnura ^oy is presented and shows fair agreement with

Straight lines ToThe daU Usïlâ íTÍJAi.0*^ ^

in eqUalÍtÍeS ,¡aVe h6“' ProVed

where ft* is the third moment of f(S-) can be written in the form 1

about the origin. The first equation

Ñ. i (9.12)

while the knowledge of the values of <j\n and gether with the assumption that the coefficient unity leads to the following approximation for

omj 0 given in Table 11, to-

skewness is of the order of

Ni *

(9.13)

The last assumption may be justified by the fact that of the exponential distribution is 2.

the coefficient of skewness

therelsTdî??^8; (5-12>and »iü. Eqs. (9.4) and (9.6) ahowa that f or J dlff®rencf because there is estimated as in inverse of nedian t ■ fa d according to the distribution function Eq. (9.3), vhile here it is oh tained as the inverse of mean 5 of » according to the roneíal Zr)!

However, since the difference between Ñ and N* is as small ^q- (9.5). it is concluded that the estimations of min and O'. n alternative methods give practically identical values. 10

The damage accumulation due to fatigue was treated first as a biem by E. Parzen. -U

as shown in by the two

renewal pro-

WADD TR 61-17? 21

¡>(z) of z = f(S°) ^s^other^terestinghp^i^2 distributi0í- -unction

flrFz(2,] " s^e’ '-'by ^ ^6017

¿[f (2)] = )1 1 + -'¿[5(G)]

where

and

£rF2(2)] =re-«F7(z)

of Fz(z) and 1:(0)]

(9.14)

dz

«^[Ñ(G)] = r e" sG Ñ(G)dC

(“r vt might^ ob^nedlfthl’^r^0^’ that the distr^utions F (z) 0f z - ) r- oí the'maxirauin be observed b7 experiment. 1 wa^h 0 - G(a) as a function of G could

fatigue’uiidëf^rando^loading s^ress-interlct^1^/Under co^itions of which at the stress-amplitudes 3- arise frí^fhaVe 10 be considered sf^cc ^ u“s arife from the intermittent apolication of H£ the avpracro r\-C r/o \ i* stress amplitudes s ^ 5 x • T.' -- "uc -1-Ii^rmii,x,ent ap A stress-interactidfacitt!.’ ^ ff? tth'aïeraee »f Hs^ f™ n 1 actor ¿o. is introduced in the for¿

mA to mi

(10.1)

cUlQ i\í. wnere

without Md1dth“intemUtJnteãpplulSnshõffatÍf: Íife at stress level -i

c^attve rale the vÄTrcIhrÄdi:/ " ' ■I5 The linear

Pi V“i = 1 (10.2)

where Nc », iS the rando» life ia therefore »dified i„ the for..,

é pi V':i = ¿ Pj ^ !Ih/».

Which reproducea resulta of rand«, fatigue teats fairly

(10.3)

well.

from Eq. (9.2) considering the failure condition G(A*) = G*, the following

WADD TR 6I-I77 22

expressions are obtained:

(a) without stress-interaction

PA íVKr

G* = $1 f( V .^ ^ (b) with stress-interaction

* PÜR * Pn%

0 = ^f ‘X1 +.+ $xf,( V il0-5) where f (S.) is a statistical stress-effect function modified for stress inter¬ action, with mean mi and standard deviation (T^*. If only the average trend is considered Eqs. (10.4) and (10.5) can be replaced by

G* = PimÄ 4 .. Pn\NR (10.6)

and

G* = p^T^ ... pnmn*NR . (10.7)

The stress-interaction factor is now defined by

w = . (10.8)

oance ^ nu0 G* = P^/N and p.m * = pim. a). =p.mi0co,G*=p , Bqs. (10.6) and (10.7) are identical witn feie linear and íhe moàifieà (^uasi- linear) rules of danage accumulation Eqs. (10.2) and (10.3) if îk and ÏÏ. are used instead of and

Ni'

The conceot of stress-interaction factors leads to that of a "fictitious" S-N diagram embodying the stress-interaction effects at all stress-amplitudes

- S . Since m^Q = l/N¿, the diagram (log 3- log itiq) has a oositive slope of the same magnitude as the (negative) slope of the (log 3 - log N) diagram, is shown in Fig. I5. To each load spectrum corresponds a straight-line fictitious (log S - log niQ ) diagram intersecting the (log S - log mg) diagram at the maximum stress-amnlitude of the spectrum . The equations of these relations are ’

m. 1, max 3i,max^ (10.9)

and

,max (10.10)

■■.'ADD TR bl-177 23

Therefore, the interaction factor

u = (s/s. (10.11)

depends on Sa anH L

». jr,. :s • * “ “ •• - - ~ (small specimens in • 24 and •12 X 10-4 for

form -- *>• - stress lnteraotion effeot oan be wuten in ^ foi^

G(A) = P¿:R f*/s \ pnNR

kl=:1 lkl + .+§1 (9.2')

ltait thMre" mean

^ = S Pi"!* »d f „ = p. *2 _

S»• .- ~ution function

L(Nr) = § G - N, RmR

O']

where

= §

1 - %mR

Pr Õ; Ro

(10.12)

and

% = mR/G = ; n * n

5 Pi"i0 = S Pi"!“!

^ro = VG = /¾ Pi<r * 2 iO

can be

(10.13)

(10.14)

p¿h^ ¡i S; vf“sa:df y - Y0' ^ íRo = vo* resuluof ^„e sets of random fatiguAgstrof ÏÆ In ^ 1?(a)-^> alloy _are plotted in the t if? versus N -orl fPeci"e;,s eir:k °l ?0?5 aluminum

listed haVe been Se m?ed staig«6 of t,. fo™ U- e^"ential ^ spectra used in the«

: 4 WALD TR 61-177

fs(Si) = h'.exp (10.15)

ybf1,6 , ®i =.V3u 3X10 si,min = Si,min/Su are stress-ratios, 3 = 82,000 psi being the ultimate tensile strength. The values of s^ , p, and h in Table 12. The maximum stresses are either ^ 3.-

are listed or • ¿n r,nr\ ■ . ; — - '•'i.max = ^»900 psi ui

’i.max = oy.^OO psi. Assuming straight line fictitious fatigue diagrams the use if Eqs. (10.11) and (10.13) vdth known values pi , hla and permits the

evaluation of >/ and ^ bv trial and error. The results are presented in Tables I3 and 14 including mi0 = m^Q .

. 2^* ^ga^er I*1 Fatigue Life under Random Loading. While the variance / °¾ . 0f fatiSue Hie under random loading can be directly evaluated using Eq. (9.6) in which (¾ : mi0 and _ ¿ri0 are replaced y (TNr , m and respectively, where mRp and ¿rR0 are known quantities, the real problem is to establish the theoretical basis on which 0-¾ can be predicted from the re¬ sults oí fatigue tests at constant stress-amplitudes. In these tests is a function of and therefore of m^, m. being related to as shown in Fig. 16. The assumption is now introduced that the relation of CTA and nu is of the form 0 0

(11.1)

as indicated by the results on 7175 aluminum alloy presented in Fig. 18 that can be fitted by Eq. (11.1) with log a' = - 1.11 and b' = 0.39? or

log tr0 = 0.397 log m0 - 1.11 . (11.2)

It is interestin fatigue tests and (11.2) adequately

to note that the points (m A , (f ) obtained from random plotted on the same diagram topen circles) suggest that Eq. represents the relation of ö^g and “AO

log ATR0 = 0.397 log - 1.11 (11.2-)

Thus the assumption seems justified that the relition between <rn* = (r/O* and mg = m /= mg uj is of .the same form ^

log cr* = O.397 log mg - 1.11 . (11.2")

Comparison of Eq. (11.1 with Eq. (11.2") in the form 0-* = a'mg*b' leads therefore to the relation 0

aADj FA 61-177 25

where (H.3) ''o =

u.' = (11.4)

^(â^^ndard deviaUon of the

^RO = !k h <0 = fs^Ao»? (11.5)

>*- the quantities p,. and = ^ are knoun_

agreement witfdirect^obí^lalues \ (+\1,5) are found to be in good statisr10!1 °f the assuraed damage mechad.sin^^q6^031 approach based on statistical rules appears to be insHffT ^ (9,1) the application of served values, values based on Eq. (11 0*aJn ^-6 three rows of Table 15,0b- the purely empirical Eq. (n 2'0 a and values obtained with the aid nf an agneement betten Séo^^Vobse'mtta^' ahows^faan fatigue; it also shows that vq (-m p»\ ° as can in general be expected in

action. UJo = %w.-. «r;0>roÄ Äi8«^

J1 aaed . ae sumvorship flmcUon .s of^eaJ^O values has been

1(M) = esp [ .(¾ L vv-:i0U (11.6)

Srv visLi T^rct7isticn ufe at =e_i N i th log (M. f ,ïale parameter inversely related to th* \ the minimum life

consät^ineS^pluSeTl1 results in a damage a-cumnlat/o arvi considering the effect of stress inf7^1^5 random life v ^ a-cuPulaLor; rule for the evaluation of ÍJ !! teractl0n’ i . / R » this rulo could onitr Kft ^ ^ tho chirsctoristi r* between 1/a. ^ -,/n 0Uicl oniy be rigorously applied if +kc 4 nor can w , 1J 1/aR Were known. Such -1 pp a 11 the relation (11.5). be den^ 3 theoretical basis siMl^'lS

and sarne^i^tha^betweeh T/T^ “f "»“ion between l/or Ooserved valuer 1 /n ' r n/i, l'ween and VD. T0 tPqt +U5 0 /Ui Rirr TO U : 1/ai (=1/2.^03(3() and 1//V 'R/ f, est thl3 assumption IfC; Pilled against ¿«4 lnd ,„4 3 (= have [„

can be obtained in Ms^e^ ^ » hough esti/ab of l/«^' 5etS

WADD TR 6I-I77 26

„ ]2, .gPrparison Between pctremal.logarithmic formal and Proposed Distribution

7Q7^1nmf T^fi‘ Il6Fi:S‘ 2?(a)‘(c) several series of fatigue test results on l-ll have been Plotted on extremal probability paper together vath the above three distributions fitted to the test results. It is quiteT

weTfmpHWÍ th^ 0,05 < L(N) < °*95 the test results can be equally well fitted by any of the three distributions and no valid distinction can be made between them. Such a distinction appears only when the rate or "risk" of damage associated with these distributions are comoared.

^ (lhe7)ViSr funfion riN) = - dlnL(N)/dN is related to L(N) according to bq. Considering, for valid comparison, the existence of a minimum life ho not only in the extremal function Eq. (11.6) but also in the logarithmic-

be^onsSered: distribution»the blowing survivorship functions must

N-N, (12.1)

for the logarithmic-normal distribution, »ith S.. = <r [log (1(-11,)1 and 1 $ , and 1,1

$r0_^ / /Sn (l22) L J ÏU N-N0)K€y ñ-njo^ (12*2)

for the proposed distribution, where £ = <rQ/ Vm . Introducing the variable

t' = (N-N0)/(N-N0) or f = (N-N0)/(V-Nq) , (12.3)

the following results are obtained with the aid of Eq. (2.7).

For the extremal function

L(t') = exp [-t,0f] ,

the risk function is

r(t') = OCt' a-1

9 (12.4)

for the logarithmic normal distribution

Kf) =

the risk function is

WADD TR 61-177 27

r(Ÿ)- _0-^34_ X ,

fa M'ïfr (12.5) N j

and for the proposed distribut ion

1.(0- ,

the risk function is

-(0-

1£^*(-br)

/ .1 - 3/z ., ~ \fa\ j- J_ / I — "t •v'i- . Li 4-t ) exp [-z ( )J (12.6)

\-i'

ilV

(12-5) and <12-6> di«^s significantiy. As with oe >1, U ”cc«aL t Lh r m“ot°nlcaUy £« the extremal distribution

tends towards a constant value 1/2^ £0^ thfprCsed’^lbulS)'“10" ^ ^

threeIdLSL\tL?^’erp"0«erdhi2nss?2™r fUnCtl0,,S f°r the

proposed'distribution^ends^owardsb^power^unction3^11^6 aSS°CÍated with the assumption is made that the stress effLt- f„n ^ ^ f N aSytnpl:otically if the stress level but also of numbpri a ? ^ 18 n0t 0nly 3 function m (n) _ ,i.m Na"-1 1 , ., number of load applications such that the averace is

oO j - a oioN and the standard deviation is ar,fN^ - /õrr K"l/2„(2b"-1^21S mg to the survivorship function L(») = ) -fö?b / f ^ad witi Eq. (9.7) when a" = 1 and b" = 1/2. ^ ^ WhlC1 18 laentical

IV. SAmY-ANALYSIS FOR COMBINED FATIGUE AND ULTIMATE LOAD FArnrp,

°£ a ttructur^)Vthe^carrylng^pacity0^!^!)!011'‘ IlK ÍÍm1 ia“-e SO that its probability of survival diminiQ' ^ pr°?resslvely reduced by fati tion characterizin3 the aCC°rdinS to the reliability fun application of an "ultimate’^load if Under the sin8le wi'.ether still vlrtnallytaffec^ b £atl”u"“ “Î CapaCU cannot sustain this load Tn fart- \-cc aIready somewhat reduced by it

and ultimate load Laut is on! « " ^ unecr conditions of developed extensive fatlsuch.™producln8 such faili an operational load intensity “ üíltivelí hLÎt^babn í“ T™“™1 ^ recurrence period) may cause final failure ^ u-i PyobabLllt:y of occurrence (she

definition,^ faiiurc'occ^ring „1fr aü elclit! HI: iS’ ^ load intensity that happens to exceed the "static" carrv’ ' exJcPtronally raí taro, unaffected by faUgue. Thus, in the «ri) par! ^ waen fatigue damage is non-existent or insignificant the f-iil C °- 3 structure feat of ultimate load failure. As fatigne^^ellí^d""“1» ^

hVeJD TR 61-177 28

reduces the static" carrying capacity of the structure,a joint failure criterion must be formulated combining fatigue with ultimate load failure in such a way as to provide for the alternative possibilities of fatigue failure under essentially operational load conditions and ultimate load failure under exceptional loads. To this end the total load spectrum is divided, somewhat arbitrarily, into two parts; one, containing the operational loads up to a limiting load intensity (S.. ) of a recurrence period about equal to the expected operational life, and consider- ed to produce fatigue damage; the other made up of the load-intensities above

vi3 j ^ (^ax) which, because of their rare occurrence during the operational life, do not significantly contribute to fatigue.

The random application of N loads belonging to the first part will produce survivorship function L'(N) characteristic of fatigue and obtainable from

random fatigue tests based on this part of the spectrum. The probability of survival L‘(N) is reduced by the danger that the occurrence of a load belong¬ ing to the second (high) part of the spectrum may cause "static" failure before the fatigue damage accumulation is extensive enough to make the probability of an actual fatigue failure under a loaj belonging to the first part of the spectrum significant. Obviously the probability of such "static" failure increases as the carrying capacity is gradually reduced by fatigue. It should, however, be kept in mind that, according to test-results, the reduction of the initial resisting area by only a few percent due to a spreading fatigue crack does not noticeably reduce the "static" carrying capacity.

The survivorship function for the combination of fatigue and ultimate load is obtained by adding the "risks" or rates of failure due to fatigue and the ultimate load, determining the associated survivorship function with the aid of Eq. (2.7). Hence

r(N) = - dlnL(N)/dN = r'(N) + Pf(N) (13.1)

where r'(N) is the rate of fatigue failures and Pf(N) the rate of ultimate failures. Since '

L'(N) = exp [- A'(N) dN] (13.2) 0 9

Eq. (13.1) becomes by integration

L(N) = L'(H) exp [-/^ Pf(N) dw] (13.3)

which is the basic equation for the determination of L(N).

Distribution of Damage. The advantage of assuming a combined physical- statistical mechanism of fatigue damage such as Eq. (0.1) or Eq. (0.2) is' rot only that a survivorship function L* (N) can be derived from it but also a distribution function of A can be established that is compatible with L»(N).

WADD TR 61-177 29

»ith nean sSStiart'iiCTlftion9'2? ÍS l”™!? by a ,,°nnal dlstribution distribution function P,, (i) of A for aVxod4.^0^ “l

F¿ (Ù , j. ^ J (14.1)

tterotheíAhLd)SfoarTm40 vüüoT fUnetl°" °f A is obtained in the form ^ distributi°n function F^(N)

(14.2) F, M r i [-^Ä]

Which turns out to bo E,. (9.3) if A is replaced by A *.

« ““~srrss“r^i:„r*u

^ Sñ is a constant and Oe(A) is a »onoto^cally i„creasi„c function

For constant value of A , the distribution of N

~ I — Cxp

Which turns out to be Eq. (11.6) when

(14.4)

is introduced. T. u G>e(A*) V-No -

fatigue mechanism fron^ich^he^i^ersion^f ti* ^ ^ (l4,4) 10 a would follow automatically. ^ 1 n °* fatlgue lives under random load

of A míewrtSSs^iídlíaLd! eiPreS3ion of the distribution

aX Mathematical Fonmil at.i An 0f í?ísi, 0f rai i,, . C/A for the history function wV/ ? Assuming the form

G(A)r a^/k'c 3:eîr k" / tv i °roTKi<niiiai conftio"5- therefore, ^ and 0 ^ < i, ^ becomeSi

X' n

A =ci?.f(v K» I (15.1)

Which implies that the trend of the relationship

WADD TR 61-177 3vh

K' , M A = c m ■ N or (15.2) Kl

is to be observed by experiment for constant or random loading, where c' = X c. For the present discussion, A is defined such that

A = (¾ - Rjj)/r^ (15.3)

where R^ and % are the initial and the momentary static strength of the structure or specimen respectively. Combining Eqs. (15.2) and (I5.3),

- R» X'

R1 c'dijN ( or = c’m^N), (15.4)

In the case of uniaxial loading RN = TQ A» and is the material strength parameter and A{j and A^ J‘are ___ initial cross-sectional areas of the structure or specimen* Inserting these into Eqs. (15.3) and (15.4), -

% ’ Y1 where Tq e'momentary and the

A =(A1-An)/A1 (15.3.)

and

A, - A X »Cn^N. (15.4»)

lhe latter relation is fairly well realised in experiments, for example those performed by W. Weibull 10 and by J.L. Kepert and A.O. Payne. 19 Using logarithmic scale for both A and N, Fig. 24 shows the experimental results, to which straight lines can be fitted. This suggests the validity of Eq. (15.41) at least as a first approximation. On the basis of this consideration, the mathematical formulation of the risk of failure r(N) = r'(N) ♦ Pf(N) can be established, emphasizing the determination of the risk of failure Pf(N) due to the load from extreme portion of the spectrum since r' (N) is derivable fron the known function L'(N).

Siting RN=R10„(A) and v’N = RN/S = ¿N(A) f^/s where S > S.-. and 0?m(A ) is a monotonically decreasing function of A , and also writing y = Ri/S, the risk of failure Pf(N) (the momentary probability of failure) is

given by either Eq. (I5.5) or Eq. (15.6) .

PfOO = p( vN < i) = p [ v' ¿N( A) < 1 ] (15.5)

Pf(N) = PfRfr S < 0) = P [¡^„(A) . S < 0] (15.6)

It is possible to obtain the distribution of A from the postulate that G(A) is normally distributed: the knowledge of this distribution in turn permits the

WADD TR 61-177 31

determination of the distribution of ¡ÉN(A).

For the evaluation of ¿t.

hZttn”g-the P«>b¿iHty^nsiT?Únhr«PlÍeí ^ (15’5) i® «eed; typo thesis that S > s by fc f/íw ^0n of S relative to the

^o”“1“10'«1 dietribSion F ^ „V of V f (= ,,here Pt = ^(W Hypothesis IS Obtained making use of r/h) y? relative to the same3 ’ of tf1 1‘ tte “"ditional distribution2 Fo1 ?v' ln the 6ane iMnner as of the product of V ami ^ (

This can^e justim? bythe^t ttat^’í1*“1 variati0" “f Ri is legieoted. narrower than that of ¢6 k,f A1 ho 3 Pbdbistical variation of Ri is nuck* the distributions of bot,“ J' ^ f’ al-pUfying ajsumot^ actual analysis extremely difficult^xcüpt 3i„°g 3 form which "okes

^ = <R1 - %)/«!,

<%(*)= 1 - A (15-7)

and

yN= ^(1-A). (15.8)

The distribution of v'-r,/q ^ directly obtained from the dístritatL'ntf ^34 R1 is “estant i,

Assuming an exponential probability density

fs(S) = h'exp [-h'ÍS-S^)] (S>W (15.9)

£ir°Ä:t:^Ä for airplane wi^

fai^,^^^rirÄ: ^ load

(ws,WÄl0)

f; (v') = exp[-h,(Rl/v. .W] (15>11)

which gives the density function,

WADD TR 61-177 32

(15.12)

h® R

f v' ( ^ = exp t'h’( V“' - w] .

By virtue of Bq. (I5.I), A itself is normally distributed with mean mRN and standard deviation O’r IN, assuming H'= 1 and c = 1. Therefore the distribution of A = 1 - A is also ittrmal with probability density

fZ(i) = ejp { -[Ã -U-SrNJ 2/(2NJ2r)| . (15.13)

Making use of Eq. (15.12) and (15.13) it can be shown that the distribution function Fv^(vn) of W*N = y,(l-A) = v*Ã is given by

FvV VM)= í&WV'rVh'<rR(|/>'V/<’'RN] exP [h'i3nax-^l/V'r) +

* 2h'‘irR/''B Rn] (15-14)

where RN = R-^l - mRN) and = ^ ßrR. Then Pf(N) is

Pf(N) = p. X f![( Smax"RN h iTp )/(Tr Rn «n

j exp fh' ^Smax”RM^ +

(15.15)

where pt = P(S > Snai) = exp [- h'(3„ax - s^)] .

When H ji 1, the best way is to apply Eq. (1.2) directly in conjunction with Eq. (15.6):

Pf(N) = J* h'exp[-h»(S- Sminj] FR (S)dS (15.I6) max N

Since G(A) = p- [(R^ - R^/R-j] is normally distributed with mean NnL and standard deviation , K

R

frn(%) = $ [ (15.17;

Substituting Eq. (15.1?) into Eq. (I5.I6), the desired expression for Pf(N) is obtained: 1

[-»'(s-Wl d3 (15.18)

WADD TR 61-177 33

i» giveVbyt! (14.1k' 'SrÄ'"'-9” (*.* = contenu), the trend of the'relLn 1^™A 6 aj =„ 17“

A'' * = dñ^N - W0)

which is essential]^ the same as E''. (15 2) Tho + -u 1 distribution of RN is

(15.19)

%(%) = 1-exp . f JdtMoÄy^Vf lí B1 * % / . (15.?o)

ds. (15.21)

Finally,

PfW'•Çmaxh'eilPl‘h'(S‘WJ 1-exp ■ 0ld (a~”°^ —

(ISO)!" ''UrVlTOrShip ^'¿»n 1(H) can now be obtained with the aid of Eq.

‘'SlaeVSlfi. etaengtTl1 W 0n on 20*

iS.' :~Virirr^SrV4~~-.i -.r^.tsr- rSionentl?í distribution. Table 16 shows thesfstr^ 7°^ rati°S drawn from 10 ratios, lhe exponential distribution of +>í ? ®tress levels and their cycle

24 h O) = 2.70110-4 exp f-2.70xl0-*(S . 19,200)^) •

f%(S) = 2-70x10''' exp ( -2.70x10-^(3 . 57,600) ] (16.1)

This experiment is understood in « u 1 for which S Is „ater than „.boota was ton °f 11,6 dist--ibution IS obtained by setting in Eq. (15 in) - truncated: in other words, fe (3)

«»«as ,1Mk ^ si WW& arpiad £.

and c = nftaaí "“"«-leal example that K'- 1 quantities in Eq. (I5.I5) are5 d‘ F°r the examPle considered the

Pt = 3.16 X IO--5 , Srnax = 57,600 psi

hf = 2.70 X IO“4 , hn = 64,000 (1 . ñR •,)

and 0"rn = 64,000 ^j(J^

5^ WADD TR 6I-I77

where mR and <TR are evaluated after the determination of G* [= GU*)].

For K ' = 1,

=* =4* = (Rt - RN»)/R1

sprrn ialls "hcn its not detemiae %* uni,„ely becaus¡ of thc aLtLucáf’oL^cÍr^'íh” í“dillg.

tribuL^rSe^go^s TalZ T! 0£,the l0ad not con- not affected by this part of the snpr? ° s^teraent t^iat “RO 411(1 Orq are Sether with the fact th - there i«/ rtUa* C°nsiderl-n8 this assumption to-

54,000 psl and that f^ A <e5Í top rSandlnda t8PeCînm ^ her of loads higher than P * I H T denoting by TR* the return num-

% iron R = 54,400 paTcau'ii by T Hi thf .de<:rease °f tha »atic strength Oq. (16.2) making use o£ Eq (15 2( Ca" be e!IPresSietl 6y

^ = (64,000 - 54^400)/64,¾ 1¾1¾ Z r/¾¾ St"ce

■ 0-l5 = \o^V (16.2)

appro=ir^tc^e^^^d^“rf|ud:---- ^ “ ‘ £i-

I?6,-rÄ --^ÄtÄ r^d o£ Eq,

V = ¿ 54,400 psi

nance the value 54,400 psi can be used as RM# or 0 1¾ as c - are, therefore, N u,i:? as A * ^ and aR

\ B °*15 \Q = 4.47 X io -8

% = 0.15 ã, RO = 3*30 X 10-5.

n» “^5! t: “Lu S-l^^ntbS^rTrl0” °£ pd») - ^ [Ä:LÜN).1s also tabulated i„ laúals^I ^Tpíg.^' 27’

iSt dVilb e 91S s“' N)yishe "“f0“ Integration and

“V *;= * -ssä ä:-:.(3u of loads pertaining to the extreme portion of thc spectrL. appllCatlon

WADD TR 61-177 35

V, BI3LI02RAPHr

1.

2.

3.

5.

6.

?.

8.

9.

10.

(,) F^thg. of Structures", Trausaotious, ^

(t> 3tracturai Faiiuren-

(e) JaU«fÄ’ of (a) ProLseuu»« à u

•lions , Revue jenerale des Chemins de Fer, June 1931.

(b) 3il,o,. P. and 5. Torroja, "Dctenrânaçion del Coefficiente de r^,,lri_

Madrid,*i950^StÍntaS InStlt“to ^ la ConstrÂ

^ ^ '’'"'t.. JJ*) < -'-rencth, Safety and Economical Dimensions of Struc-

S.^ttÂInrt‘ ,f ^6nadS5tatÍI<' ^ mat, ,f

ms?«« asirs»*'M* - (g) "Fin^ Third International Cong, of Bridge and Structural Eng.

liege, ^elguuin, pp. 635-636, 1948. s ’

(f) n?T^lZÎ RTrr’/°UI? International Cong, of Bridge and Struc- ■nual Eng., Cambridge, England, pp. I65-I76, I95,?.

(C) "FÍ¡W R3?rllhl ínternational Cong. of Bridge and Structural Eng., Cambridge, England, pp. I45-I65, I952.

Freudenthal, A.«., "Methods of Safety Analysis of Highnay Bridges» Frei manary Publications, Sixth International Gong. ofSidge ' and Structural Eng., Stockholm, s»eden, pp. 655-661., i960.

Huntington, E.Ï., "Frequency Distribution of Product and Quotient" Ann. Math, statistics, Vol. 10, p. 125, I939, " ’

Cumbel, F.J »Statistics of Extremes», Columbia University Press Jew York, p. 20, I958. 7 rress,

Gunbel, E.J., ibid, p. 2?3.

Gumbel, E.J., ibid, p. 2?0.

Freudenthal, A.M., loc. cit. ref. 1(b), p. I359.

Freudenthal, A.M., ibid, p. I358.

American Concrete Institute, "Building Code Requirements for Re<n- forced Conorete (ACI 318-56). A 6CH. Load Factors".

WADD TR 61-177 36

11. -’eibull, W., "A Statistical Theory of the Strength of Materials", J.V.A:s Handl No. I51, p. 29, 1939.

12. Smith, W.L., "Renewal Theory and Its Ramifications". J, Royal Stat, Soc. Series 3. Vol. 20, No. 2, pn. 243-2%, 1953.

13. Parzen, E., "On Models for the Probability of Fatigue Failure of a Structure”. Advisory Group for Aeronautical Research and Development, Report 245, April 1959.

14. Freudenthal, A.K. and R.A. Heller, "On Stress Interaction in Fatigue and a Cumulative Damage Rule». J. Aero/Space Sciences, Vol. 26, No. 7. July 1959.

15. Hardrath, H.F. and C. Utley, Jr., "An Experimental Investigation of the Behavior of 24 5-T4 Aluminum Alloy Subjected to Repeated Stresses of Constant and Varying Amplitudes", NACA Technical Note 2?98, October 1952.

16. Freudenthaï, A.M, and R.A. Heller, "On Stress Interaction in Fatigue and a Cumulative Damage Rule. Part II. 7075 Aluminum Alloy". WADC Technical Report 58-69, Part II. January I960.

17. Freudenthal, A.M. and E.J. Gunibel, "Physical and Statistical Aspects of Fatigue", Advances in Applied Mechanics, Vol. IV, Academic Press Inc., New York, N.Y., 1956.

18. Weibull, W., "The Propagation of Fatigue Cracks in Light Alloy Plates", SAAB Technical Note 25, 1954.

19. Kepert, J.L. and A.O. Payne, "Interim Report on Fatigue Characteristics of typical Metal Wing", NACA Technical memorandom No. 1397, March I956.

WADD TR 61-177 37

TABLE 1

Relation between Standard Deviation áR( ¿s) of

log R(logS) and Coefficient of Variation v R( vg)

based on mean R(5), or based or, Median

K(S) of R(S) for Logarithmic Noma Distribution

T (T) 0,05 0a° 0a5 0.20 0.30

r( g) 0.0217 0.0431 0.0641 0.0844 0.123

vR(vs) 0.0499 0.0995 0.1½ 0.I96 0.288

TABLE 2

Standard Deviation of ¿y? and O’Jr

of R and S. R

log R t log S as a Function for Logarithmic Normal Distribution

WADD TR 61-177 58

TABLE 3

ReUUon between Probability of Failure Pf and Central Safety Factor v 0 for logarithmic Normal Distributions of R and S

WADD TR 61-17? 39

TABLE 4

Relation between Parameter ä(/3) and Coefficient of Variation vR( Vc) based oiHean R(S) or <rR/?( 43)S on Characteristic Values H(3) for Extremal Distribution of R(S)

(a)

0.05 0.10 0.15

O.O5II 0.104 0.160

24.4 11,8 7.41

(b)

0.10 0.20 0.30

0.956 0.183 0.264

7.84 5.75

Vs VS

ß

WADD TR 61-177 40

TABLE 5

Relation between Probability of Failure Pf and Central Safety Factor y q for Extremal Distributions of R and S

£s £b _1_

3 R 10_1 10*2 10“^ 10‘4 10“5 in“6

o 0.05 1.10 1.20 1.33 1.45 1.40 1.77

0 0.10 1.20 1.50 1.80 2.17 2.60 3.20

0 0.1S 1.30 1.-85 2.60 3.55 MO 6.45

0.10 0.05 1.20 I.50 1.75 2.00 2.3O 2.75

0.10 ojo 1.30 1.70 2.05 2.50 3.00 3.70

0.10 0.15 1.4S 2.00 2.7S 3.80 5.20 6.00

0.20 0.05 1.40 1.80 2.50 3*35 4.40 6.00

0.20 0.10 1.45 2.00 2.70 3.60 4.90 6.90

0.20 0.1S 1.60 2.30 3.40 4.70 6.40 8.^5

0.30 0.05 1.50 2.30 3.50 5.20 7.70 11.30

OJO 0.10 1.60 2.45 3.70 5.50 8.20 11.80

0.30 0.1^ 1.70 2.70 4.2^ 6.35 9.5J 14.00

WADD TH 6l-i77 41

table 6

Ratios and Eq for logarithm! o Normal Distributions

(a) r P

(b) s q

WfiDD Ttí 6I-I77 b2

TABLE ?

Ratios Tp and for Extremal Distributions

(a) rp

VR 0*°5 0.10 0.15

<* 24.4 11.8 7.41

r0>1 0.912 0.826 0.738

(h) Sq

frg/S 0.10 0.20 O.3O

P 14.3 7.84 5.75

s0.1 1*17

so.oi

s0.001 1,62

so.oooi

1.33 1.48

1.80 2.23

2.41 3.32

3.24 4.96

WADD r-i 61-177

T4BIÆ 3

Improvement of Matpri ai as a Funetiop of n ^o vis °f Decrease o£

of Failure pf - i0-o of „ " ;'n^ure Constant Probability Members ( Vg = 5.0) l0n"aedu^ant Structure of n

TABLE 9

Increase of Central Safety Factor Vq as Function of n to Ensure Constant Probability of Failure Pf = 10“6 of a Non-Redundant Structure of n Members

n 1 10 20 50 100 200 500 1000

R 10“6 5^0-72^0-¾-7 5x10"82x10"810’8 5x10"92x10‘910“!?

Pk

_s ?

0 0.05

0 0.10

0 0.15

0.10 0.05

0.10 0.10

o.io 0.15

0.20 0.05

0.20 0.10

0.20 0.15

0.30 0.05

0.30 o.io

0.30 0.15

1.2? 1.28

1.60 1.62

2.01 2.06

1.69 1.72

1.95 1.99

2.32 2.38

2.59 2.66

2.82 2.90

3.18 3.29

3.90 4.06

4.14 4.32

4.53 4.75

1.29 I.30

I.65 1.68

2.11 2.I5

1.76 1.78

2.04 2.08

2.46 2.52

2.77 2.84

3.03 3.11

3.45 3.56

4.38 4.45

4.57 4.75

5.04 5.25

I.3I I.32

I.70 I.72

2.20 2.25

1.81 1.84

2.12 2.16

2.58 2.66

2.92 3.01

3.20 3.32

3.68 3.82

4.61 4.83

4.94 5.18

5.48 5.76

I.32 1.33

I.74 I.76

2.29 2.33

1.86 1.89

2.20 2.23

2.7I 2.77

3.03 3.I5

3.40 3.49

3.92 4.04

4.98 5.16

5.36 5.55

5.97 6.20

I.34 1.35

1.79 1.81

2.3q 2.42

I.92 1.95

2.’8 2.32

2.85 2.90

3.26 3.33

3.61 3.70

4.20 4.32

5.40 5-58

5.82 6.02

6.52 6.75

10,000

lo-io

I.37

1.88

2.55

2.01

2.44

3.10

3.58

4.0]

4.73

6.20

6.72

7.61

WADD TR 61-177 45

TABLE 10

i:’te load Factors ci and „ of Probability of Failure^ 35

JUL JiO.

1.0? 1.34

0.01 0.991

J-.o/ 300x10- 2.02x10-2 1-90x10., 1<00id0.5 ^omo_7

0.01 0.685

0.997

0.912

1.004

1.138

1.008

1.251

1.011

1.365

(b) Voi ** 4.01 0-=0.5)

_rl.'

1.14

;r .6,8^0^ 306x10-5 ;ai0-6 1.060

1.385

WADD TR 61-17?

TABLE 11

and Standard Deviations <Ji0, of Stress Effect Function

Test No.

Stress* Level

ni0

ffi0

1

const. 29,500

1.48x10-?

1.6lxl0"4

2

const. 37,300

7.87x10-7

3.04x10-4

3

const. 44,900

4.31x10“6

6.20xl0“4

4

const. 52,600

1.64x10"^

1.16x10-3

5

const. 60,300

4.65xl0“3

1.00xl0"3

Test No. Z

Stress const. Level 68,000

»lo'^o 2-71ld0'3

20

random

l.llxlO"6

3.99xl0-/+

21

random

5.49x10"6

5.05X10’4

22

random

2.21xl0‘5

8.64xl0"4

23

random

8.34x10*7

4.06xl0-4

Test No.

Stress Level

mR0

24

random

4.97x10"6

6.l0xl0"4

25

random

2.03x10*3

8.68xl0-4

26

random

1.35x1o"7

2.03xl0‘4

27

random

3.39x10*6

6.71xl0"4

28

random

1.25xl0*5

9.47x10*4

* Stresses are given in psi.

WADD TR 61-177 4?

TABLE 12

laarJ Vectra for Random Fatigue Tests of AA 70?5 Al.

Test s Series i .35 ,45 .55 to. h___

20 I7.3 .822 .1456 .02664

21 I7.3 .822 .1456

22, 17.3 _.822

23 22.9 .900 .0900 .00900

24 22.9 .900 .0900

25 22.9_.900

26 34.3 .9684 .0306 .00100

27 34.3 .9684 .0306

28 84.3_.9684

.65 .75 .85 .95

.OO453 .00100 .000182

.02664 .00458 .00100 .000182

.1456 . 02664 . 00453_.00100

.000900 .0000900 .0000100

.00900 .000900 .0000900 .0000100

.0900 .00900 .000900 .0000900

.000030 .000001 .00000003

.00100 .000030 .00000100 .00000003

.030600 .00100 .0000800 .00000100

TABLE 13

Inverse of the Slope, yj , of log S - log üIq Diagram

Test " --

J&X 20-21 22 23 24 25 26 27 28

>| 6.74 6.68 6.14 6.46 6.32 5.38 8.24 6.56 6.54

WADD FR 61-177 46

I

TABLE 14

Stress Interaction Factor U). , Mean m*i0 and Standard Deviation cyi0 (in Terms of G*) of Modified Stress Effect Function.

Test Stress No. level S.,

Si

dV^n2) 2.8?xl04

itUg 1.12x10“^

(Ti0 1.35xl0-4

20 p. 0.822

ici 3.79

4.23x10“^

3.69x104 4.51x104 5.?3xl04 6.l5xl04 6.9?xl04 ?.79xl04

8.83x1o“'7 4.60x10"^ 1.91x10-5 5.96xl0-5 1.66x10"4 4.12xl0“4

3.0?xl0-4 5.90x10-^ 1.04x10-3 1.63x10-3 2.45x10-3 3.52x10-3

0.1456 0.02664 O.OO458 0.001 0.000182

2.59 1.92 I.50 1.20 1.00

2.29xl0“6 8.79xl0“4 2.86x10-5 7.17x10"5 l.66x!0"4

<y io 2.29x10“4 4.48xl0“4 7.64x10“4 1.22x10-3 1.75x10-3 2.46xl0_3

21 Pi 0.822 O.I456 0.02664 0.00453 0.001 0.000182

wi 3.20 2.35 1.85 1.45 1.19 1.00

m*i0 2.82x10-6 1.08x10-5 3.44x10-5 3.65x10-5 1.98xl0“4 4.12xl0"4

<T*i0 4.87xl0-4 3.28vl0-4 1.31x1o"3 1.89xl0"3 2.63xl0"3 3.52xl0"3

0.822 0.1456 0.02664 0.09453 0.091

3*16 2.21 I.65 I.27 1.00

1.45x10"5 4.21x10"5 9.84x10-5 a.lOxlO“4 6.12xl9“4

9.31x10 1 1.42x10-3 1.99x10-3 2.69x1 ”3 3.‘¡2xlO"3

23 Pi 0.9 0.09 0.009 0.0009 o.oooo) 0.000009

4.73 3.06 2.14 1.61 1.24 1.00

ra*i0 5.32x10-3 2.71xl0"6 9.34x1o"6 3.06xl0"5 7.38x10-5 1.66xl0“4

O*i0 2.51x1o"4 4,79xl0“4 7.93x1o"4 1.25X1.0“3 I.7 10“3 2.45xl0"3

D. ‘ 1

i-i

* n

iO *

O’ ; iO

WADD T9 6]-177 49

Table 14 (Cont'd)

Test No.

S1 33 s,.

24

m iO

°-9 0.09 0.009 0.0009 0.00009 0.000009

4,19 2*86 2.06 1.59 1.24 1.00

3.70x10 6 1.32x10-5 3.94x10-5 9.45x1o"5 2.0òxl0"4 4.12x1o-4

5.41x1o-4 S.96x10-4 1.38x10-3 1.96x10-3 2.67x10-3 3.52x10-3

«oi *

ra iO

iO

0,9 ú-09 0*°09 0.0009 O.OOOO9

3-65 2.44 i.76 i.31 L00

1.67x10-5 4.64x10-5 1.05x1o"4 2.17x10-4 4.12xl0"4

9.87x10 1.48x10 5 2.04xl0"5 2.72xlo"3 3.52xl0“3

26 p. 0.9684 0.0306 0.001 0.00003 0.000001 0.00000003

Ui 0.955 0.969 0.977 0.986 0.993 1.00

m^i0 1.06x10"' 8.55x1o-' 4.49x1o-6 1.88xl0"5 5.92xlo"5 1.66xl0"4

(T ip 1.32x10 3.03x10 4 5.85x10"4 1.03xl0-3 1.63x10"3 2.45x10

27 n. ‘ 1

<0

m iO

U)

m 10

<^10

0.9684 0.0306 0.00! 0.00003 0.00000! 0.00000003

3,50 2,51 i'8? 1.50 1.21 Loo

3.09x1o-6 1.15x10-5 3.60XX0-5 8.91x10-5 s.oixlo-6 4.12xl0-í(

5.05x10-^ B.ÍZM-* !.^q-3 1,92x10-5 2.69x10-3

0.9684 0.0306 0.001 0.00003 0.000001

Z-5U 1-50 1.21 i.oo

1.17x10"^ 3.62x10-5 8. 15x10-5 2.01x10-'* 4.12x10"'*

3.55x10-** 1.34x10-3 1.92x10-3 2.64x10-3 3.52x1o'3

WÆDD TR 61-177 50

TABLE 15

Comparison of Standard Deviation 5 Rn Estimated by Various Methods

Test No. 20 21 22 23 24

Experiment 3.99x10“^ 5.05^0-^ 8.64x10"^ 4.06x10"^ 6.10x10"^

Theory 3.14xl0“4 6.05xl0"4 1.08xl0-3 2.84xl0"4 5.97xl0*4

Eq.(11-2") 3.35xl0‘4 6.32xl0-4 l.lOxlO“3 3.08xl0"4 6.08xl0"4

Test to. 25 26 2? 28

Experiment 8.68xl0“4 2.03xl0“4 6.71xl0‘4 9.47xl0~4

Theory 1.06x10“^ 1.44xl0"4 5.20xl0"4 8.75xl0“4

Eq. (U-2n) 1.0?xl0“3 1.^6xl0”4 5.23xl0-4 8.85xl0-4

WADD TR 61-177 51

TAELS 16

Load Spectrum for Random Fatigue Test of AA 2024 Al.

Stress Level S, 5 s o _1 2 3 S4 S5 J6

S in su 0.35 0.45 0.55 0.65 0.75 0.85

3 in psi 22,400 23,800 35,200 41,600 48,000 54,400 60,800

Cycle Ratio .822 .1456 .02664 .00458 .00100 .000182 0

TABLE I?

Probability of Failure Pf(N) as a Function of N

52 WADD TK 61-177

TABLE 18

Survivorship Function associated with Fatigue

L‘(N)

N 103 104 lo5 2xL05 4xl05 6xl05 8xl05

L*(N) 1 1 1 1 1 1-1.51x1o'-5 0.9996

log L‘(N) 0 0 0 0 0 0 = 0

N 106 2xl06 4xl06 6xl06 8xl06 107 2x10^

L'(N) 0.99720.86860.352 0.102 O.O2740.OO678 5.37xl0‘6

log L'(N) 1.998 1.939 Î.546 1.009 2.438 3.831 6.730

TABLE 19

Survivorship Function for the Combination of Fatigue and Ultimate Load

exp [- /0’'Pf(N) dN]

N 104 2xl04 4xl04 6xl04 8xl04 105

exp £-/^Pf(N)dN ) 0.?44 0.893 0.798 0.700 O.615 0.549

N 2x105 4x105 6x105 8x105 106

exp f-/WPjN)dlO 0.283 O.O6O3 0.00912 0.000912 O.OOOO63I L 0 f

WADD TR 61-177 53

% WADD TR 61-177

with Logarithmic-Normal Distribution

WADD TR 61-177 55

Figure 1.

Distribution Function of

with Logarithmic-Normal Distribution of R and S

MDD TR 61-177 56

Fig. 1 (c)

Figure 1.

Distribution Function of

with Ingarittaic-Nornal Distribution of R and S

UADD TR 6I-I77 57

figure 1.

Distribution Function of

WADD TR 61-177 59

Figure 1.

Distribution Function of

with '

Jgarithnic-Nornal Distribution of R end S

'WDD TR 6I-I77 60

WADD TR 61-177 62

—-(Od ï UADD TR 61-177 63

'..'AJJ TR 0I-I77 C

00

'."Oü TR 61-177

\.;jd tu 61-177 67

o .

. Fig. 3 (d)

risore 3.

Distribution Function of

with Extremal Distribution of R and S

i

I

6 Q ö

‘.JD IR 61-177

'....JD tí; 61-177 67

. ï. . TR 61-177 b9

CO

UADD TR 61-177 70

3.

Distribution Function of ^fth^rtrenai Distribution of R and

0

•• —

^0 K "

b >*

0 W

l«n Ò V ■

M •> b »

O >

S </>

0

6 J

CO

'S et)

ai

On O

s O

■H +3

£> •H (h

+3 n

O) Ch

M •H Pc«

<Vh O

§ •H ■P O

Ci O

•H +3 XI •H C« 43 m

©

&«

•o • o lO iO -('Md

U«\ÛD TR 61-177 71

WADD TR 6I-I77 72

with Extremal Distribution of R and S

w

1 « «H O m §

•rl ■P

•H

•P m

£

m 5

<c> ir» ó «v

(*)d °

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WADD TR 61-177 73

Fig. 5 (a)

-« «VS'O.o^/R.O.IO

butions of R and S

Wj\DD TR 61-177 74

-e <r8/§*O.IO, <rH/fi = 0.05

-^- (P'O.I, q = 0.000l)PN

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Fig. 5 (d)

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0 0.5 i is

Fig. 5 (e) Figure 5. Relation between Probability of Failure Pf and

Central Safety Factor q and "Conventional" Safety Factor with Logarithmic-Normal Distri¬ butions of R and S

76 V1ADD TR 61-177

Fig. 5 (D

ct-j/S '0.20, <rR/R = 0.05

Fig. 5 (g) Figure 5. Relation between Probability of Failure P/. and

Central Safety Factor q and "Conventional" Safety Factor with Logarithmic-Normal Distri¬ butions of R and S

■ADD TR 61-177 77

Fig. 5 (h)

(Ts /S =0.20, <rR /R =0.15

u ,v

Fig. 5 (i) Figure 5. Relation between Probability of Failure Pf and

Central Safety Factor 0 and "Conventional" Safety Factor with Logarithmic-Normal Distri¬ butions of R and S

UADD TR 61-177 76

Fig. 5 (j)

o-s/S = 0.30, aR /R*O.IO

Fig. 5 (k) Figure 5. Relation between Probability of Failure and

Central Safety Factor q and "Conventional" Safety Factor with Logarithmic-Normal Distri¬ butions of R and S

UADD TR 61-177 79

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Fig. 6 (b) Figure 6. Relation between Probability of Failure Pf and

Central Safety Factor vq and "Conventional" Safety Factor v" with Extremal Distributions of R and S

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UADD TR 61-177 Ö2

Figure 6. Relation between Probability of Failure P^

» and

Central Safety Factor

0 an<i "Conventional"

Safety Factor

with Extremal Distributions

<rs/S=O.IO, /R =0.05

Fig. 6 (d)

0s/S =0 10, 0r/R =0 10

Fig. 6 (e) Figure 6, Relation between Probability of Failure Pf and

Central Safety Factor q and "Conventional" Safety Factor with Extremal Distributions of R and S

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Fig. 6 (h) Fii^xre 6, Relation between Probability of Failure Pr and

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Figure 6. Relation between Probability of Failure Pr. Central Safety Factor and "Conventional" Safety Factor with Extremal Distributions of R and S

and

WADD TR 61-177 B6

crs/S*0 30, o’r/R*O.IO

Fig. 6 (k)

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Fig. 6 (l)

Figure 6. Relation between Probability of Failure Pf and Central Safety Factor „ and "Conventional" Safety Factor with Extremal Distributions of R and S

■VDD TR 61-177 87

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WADD TR 61-177 90

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Figure 13. Separate Load Factors a and>9 as Functions of Probability of Failure 4 q J

WADD TR 61-177 91

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Figure IL. Representation of Constant Amplitude Fatigue Test Results on AA 7075 Aluminum in /Ñt

Coordinate System

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Figure 16. S - m0 Relation lor AA 707i> Al.

WADD TR 6I-I77 94

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Figure 17- «^presentation of Random Fatigue Test Results on AA 7075 Aluminum in

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IN PERCENT

Figure 2li. Empirical Relation between Reduction of Static Strength and Cycle Ratio (W. Weibulll° and J. L. Kepert and A. 0. Payne^-9)

WADD TR 61-177 10h

STRESS RATIO 0.3 0.4 05 0.6 0.7 08 09 1.0 STRESS 19.2 25.6 320 38.4 44.8 51.2 576 64.0

(Ksi) 224 288 35.2 41.6 48.0 544

Figure 2^. Schematical Representation of Exponential Truncated and Full Load Spectrum

WADD TR 61-177 105

Figure 2f). Representation of a Random Fatigue Test Result on AA 202I4 Aluminum in ^Nt - N Cooruinate System

106 WADD TR 61-177

107

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Figure 27. Probability of Failure Pf(N) due to Extremes of Load Spectrum

as a Function of Number N of Load Application

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