Optimization of Time-temperature Parameters

N A S A T E C H N I C A L NOTE

m h 04 N I

n z c 4 m 4 z

NASA TN D-2975

4 . / -

-4 - 0 "

OPTIMIZATION OF TIME-TEMPERATURE PARAMETERS FOR CREEP AND STRESS RUPTURE, WITH APPLICATION TO DATA FROM GERMAN COOPERATIVE LONG-TIME CREEP PROGRAM

by A Zexunder M endelson, Ernest Roberts, Jr.? QndS . S. Munson

Lewis Reseurch Center CZeueZund, Ohio

NATIONAL AERONAUTICS A N D SPACE A D M I N I S T R A T I O N WASHINGTON, D. C. 0 AUGUST 1965

k

OPTIMIZATION O F TIME-TEMPERATURE PARAMETERS FOR CREEP

AND STRESS RUPTURE, WITH APPLICATION TO DATA FROM

GERMAN COOPERATIVE LONG-TIME CREEP PROGRAM

By Alexander Mendelson, Ernes t Roberts, Jr., and S. S. Manson

Lewis Research Center Cleveland, Ohio

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $2.00

OPTIMIZATION OF TIME-TEMPERATURE PARAJMETERS FOR CREEP AND STRESS RUPTURE, WITH

APPLICATION TO DATA FROM GEFWAPT COOPERATIVE LONGTIME CREEP PROGRAM

by Alexander Mendelson, Ernest Roberts, Jr., and S. S. Manson

Lewis Research Center

SUMMARY

By t h e use of orthogonal polynomials developed f o r d i sc re t e s e t s of data, t h e least-squares equations f o r determining the optimized s t ress - rupture para- metr ic constants a r e obtained i n nearly uncoupled form; thus t h e use of high- degree polynominals i s permitted without t h e l o s s of s ign i f i can t f igures . Optimum values of t h e constants can thereby be accurately obtained. The method i s applied t o t h e da ta obtained from t h e German cooperative long-time creep program by using a general parameter of which t h e Manson-Haferd and Larson- Mi l le r parameters a re spec ia l cases. Good cor re la t ion w a s obtained. An analy- s is i s a l s o made of creep da ta obtained for columbium a l loy FS-85 with good r e su l t s . A complete Fortran I V computer program i s included t o a i d those wish- ing t o use t h e method.

INTRODUCTION

One method of extrapolat ing short-t ime creep-rupture da ta t o pred ic t long- time l i f e involves the use of a time-temperature parameter. This concept i s based on t h e assumption t h a t all creep-rupture da ta f o r a given mater ia l can be cor re la ted t o produce a s ing le "master curve" wherein t h e stress ( o r log stress) i s p lo t t ed against a parameter involving a combination of time and temperature. Extrapolation t o long times can then be obtained from t h i s master curve, which can presumably be constructed by using only short-t ime data. Three well-known parametric methods are t h e Larson-Miller, Manson-Haferd, and Dorn parameters (refs. 1 t o 3). These parametric methods have t h e g rea t advantage, a t l e a s t i n theory, of requiring only a r e l a t i v e l y s m a l l amount of da ta t o e s t ab l i sh t h e required master curve.

More recent ly a general creep-rupture parameter w a s introduced by one of t he authors ( r e f . 4) t h a t includes most of t h e cur ren t ly used parameters as spec ia l cases. The ana lys i s i n t h e present paper i s therefore based on this general parameter.

A significant advance in the practical application of the parametric methods was the development of an objective least-squares method for determining the optimum values of the parametric constants without plotting and cross- plotting the data and without the use of judgment on the part of the analyst (ref. 5). This least-squares method involves, however, several practical dif- ficulties that arise from the fact that in fitting the master curve by a polynomial, the set of linear algebraic equations for the coefficients (the normal equations) are very ill-conditioned. shown to be related to the Klbert determinant (ref. 6), which rapidly ap- proaches zero as its order increases. Thus for polynomials above the second degree, it is necessary to use double-precision arithmetic (16 significant digits or more) on the computer, and for the fifth degree and above the results become uncertain even with double-precision arithmetic. This difficulty is inherent in the normal least-squares equations and is not limited only to the stress -rupture problem.

The determinant of these equations can be

The present report presents a method for avoiding the above difficulty by using orthogonal polynomials in the representation of the master curve (appendix A). The use of orthogonal polynomials for representing discrete sets of unequally spaced data is described in reference 6 and in more detail in reference 7. A further improvement can be obtained by performing a linear transformation on the stresses (or the logs of the stresses) so that all the values of stress (or log stress) lie between 2 and -2, as recommended in reference 7. As a result of these innovations, it became possible to perform all the compu- tations in single-precision arithmetic (eight significant digits) up to 18th degree polynomials without appreciable round-off error.

In addition, this report contains a complete analysis, in which the general parameter was used, of all the data for three steels that were obtained by NASA through the cooperation of Dr. K. Richard of Faberwerke Hoechst in Frankfurt and that were investigated in a long-time cooperative creep program in Germany. Some of the data from the latter investigation are included in this paper.

Finally it is shown by means of a concrete example how the parameter tech- niques can be applied to creep data to predict long-time creep. For this pur- pose the data for columbium alloy FS-85, as reported in reference 8, are used.

A complete Fortran IV program, as used on the IBM 7094 computer in making the calculations, is presented in appendix B. This program can be used for the objective analysis of any set of creep-rupture data by the Larson-Miller, Manson-Haferd, or the more general parameter of reference 4.

SYMBOLS

A, B linear transformation coefficients

a,b,c elements of coefficient matrix

D standard deviation

2

K

m

n

p( 0 )

Q

(2

r

S

T

Ta

t

t a

U

X

X

Y

Y a

a, P

(5

z

degree of freedom

degree of polynomial

number of data points

creep-rupture parameter

polynomial

s t r e s s exponent

temperature exponent

sum of squares of res idua ls

temperature

temperature in t e rcep t

time t o rupture

time in te rcept

coef f ic ien t of polynomial function

scaled log s t r e s s

log s t r e s s

log time

log time in te rcept

constants from recurrence r e l a t i o n

s t r e s s

o'(T - Ta)r

Subscripts :

m a x m a x i m u m

min minimum

PROCEDURE

GeneraJ. Parameter

The general creep-rupture parameter introduced i n reference 4 has t h e f o l -

3

lowing form

log t - - lnp. t a 0 -- I

where Ta, log ta, q, and r a r e mater ia l constants t o be determined from t h e avai lable experimental data. The parameter P(o) i s a function of t he s t r e s s and, when p lo t t ed against stress, i s re fer red t o as a master curve ( f ig . 1, p. 9 ) . If q = 0 and r = 1, the Manson-Haferd parameter i s obtained. If q = 0, r = -1, and q = 1 and r = 1, t h e stress-modified parameter suggested i n reference 9 i s obtained. Finally, i f q = 0, equation (1) reduces t o t h e parameter proposed by Manson and Brown ( r e f . 10).

Ta = -460' F, t h e Larson-Miller parameter r e su l t s . If

The object i s t o f ind t h e bes t values of t h e constants q,, l og ta, Ta, and r so t h a t t h e master curve bes t f i t s t h e data. To f i n d these values, t h e method of l e a s t squares i s used whereby t h e master curve i s represented by a polynomial i n t h e logarithm of t h e s t r e s s , and t h e bes t f i t i s obtained by mini- mizing t h e sum of t h e squares of t h e deviat ions ( the res idua ls ) of t h e da t a from t h e curve. The ca lcu la t ion procedure w i l l now be described. The d e t a i l s of t he der ivat ion are given i n appendix A, and a Fortran I V computer program using t h i s method i s given i n appendix B.

Calculation Procedure

To simplify t h e notation, t h e following symbols a r e introduced:

( 3 )

I T E o ~ ( T - Ta)r

y = log t

x = log 0

ya E log t a

Then from equation (1) it follows t h a t

Y = o%a + TQ(X)

where i n reference 5, Q(x) w a s represented by a simple polynomial of t he form

Q ( X ) = + alx + a2x2 + . . . + a,mxm (4)

The least-squares equations obtained sometimes l e d t o d i f f i c u l t i e s as indicated i n t h e INTRODUCTION. These d i f f i c u l t i e s can be avoided, however, by rewri t ing equation (4) i n terms of polynomials t h a t are orthogonal over t he s e t of data , as defined i n appendix A. Thus assume

4

where u j is an unknown constant, m i s t h e degree of t h e highest degree polynomial, and Q.(x) i s a polynomial of degree j - 1 t h a t s a t i s f i e s t h e orthogonality conditions described i n appendix A. The use of orthogonal polynomials permits t h e so lu t ion of t h e least-squares equations d i r e c t l y i n closed form, thus t h e loss of a l a r g e number of s ign i f i can t d i g i t s i s avoided. method of ca lcu la t ing Q j w i l l be discussed i n appendix A.

m + 5 unknown constants r e s u l t s f o r t h e case of t h e general parameter. For t he case of t h e l i n e a r parameter t he re are m + 3 constants, and f o r t h e Larson-Miller parameter t h e r e are m + 2. It i s necessary t h a t t h e number of da t a poin ts n always equals or exceeds t h e nurhber of unknown constants.

J

The

If equation (5) i s subs t i t u t ed i n t o equation (3), an equation with

The constants a r e determined so t h a t equation (3) f i t s t h e da ta bes t i n t h e least-squares sense. To accomplish t h i s , t he sum of t h e squares of t he deviat ions i s minimized; t h a t i s ,

n

i s made a minimum. Because t h e equations a re nonlinear i n some of t h e unknown constants a t r i a l and e r r o r procedure must be used. A set of values i s assumed f o r q, r, and Tat and t h e corresponding b e s t values of ya and u j a r e determined. A d i f f e ren t s e t of values f o r q, r, and Ta i s then chosen, and again the b e s t values of ya and u j a r e calculated. Several s e t s of values of q, r, and Ta a r e t r i e d , and t h e values corresponding t o t h e ove ra l l bes t f i t a r e determined. For t h e case of t h e l i n e a r parameter, only the value of Ta i s var ied (q i s always equal t o zero, and r i s always equal t o 1). For t h e Larson-Miller parameter, Ta i s equal t o -460' F, and no t r ia l and e r r o r procedure i s needed.

As a measure of t h e f i t , t h e standard deviat ion D, defined by

i s used, where K equals

m + 5 general parameter

m + 3 l i n e a r parameter

m + 2 Larson-Miller parameter

The smallest value of D w i l l correspond to t h e b e s t f i t .

5

1 . . . . .. .. .. ._ . . . . ._ - . .. .. . . .. _. - - . . - . . . . - .. .- . . . . ._. . . ..._... .... .-....----.-

To determine t h e b e s t values of ya and U j f o r a given s e t of values of T,, q, and r, t h e following ca lcu la t ions are made. F i r s t , t h e logarithms of t h e s t r e s s e s a r e sca led so t h a t they l i e i n t h e range -2 t o 2, as suggested i n reference 7. The reason f o r t h i s i s discussed i n appendix A. Thus def ine a va r i ab le X by

X = A x + B ( 9 4

4 %ax - %in

A =

%ax + xmin %ax - %in

B = -2

The polynomials Q j ( q ) a r e now ca lcu la ted f o r each of t h e data poin ts by using t h e following formulas:

- Pj -

n

i=l j n

a =

i=l

n

i=l m

i=l

Q1

where n i s t h e number of da t a points, Y i s t h e sca led value of l og f o r t h e ith d a t a point, and ~i i s equal t o &(Ti - Ta)r f o r t h e ith data point f o r t h e chosen values of Ta, q, and r.

It i s t o be noted t h a t t h e degree of t h e polynomial Q(x) of equation (5) can be increased by merely computing t h e next polynomial i n t h e s e r i e s

without having t o recompute any of t h e previous ones. This i s one of t h e advantages of using orthogonal polynomials.

%+2

Once t h e values of Q j have been computed f o r each of t h e da t a points, ya and u j can be ca lcu la ted as follows:

6

I

Let

n

i=l

n co = c a;yi

i=l

where j = 1,2 . . . m + 1.

Then

j=1

J Note t h a t i f q = 0, a. equals t he number of da ta points n. Thus by means of equations ( 9 ) t o (12), t h e bes t values of Y a and U j t o f i t t h e da ta are found f o r a given choice of Ta, q, and r. The Fortran I V program described i n appendix B automatically scans all the desired values of Ta, q, and r and chooses t h e bes t set from all t h e submitted values as determined by t h e s m a l - l e s t value of t h e standard deviation D, as defined by equation ( 7 ) . The method can be i l l u s t r a t e d by a simple example: consider a s e t of t heo re t i ca l data , which f i t t h e following equation exact ly

7

9'5 - log = 7.02 + 0.467 x + 0.061 x2 + 0.00928 x3) T - 600

Eight da t a points s a t i s f y i n g t h i s equation are given i n columns 2 t o 6 of t a b l e I. For this d a t a Ta = 600' F and log t a = Y a = 9.5. Suppose, however, t h a t t hese e igh t d a t a poin ts were obtained experimentally and t h a t t h e values of Ta and l o g ta were not known. The problem then i s t o f i n d t h e b e s t values of Ta and log t a t o f i t t h e d a t a by t h e l i n e a r parameter. These values can r ead i ly be found by using t h e equations of t h e previous sect ion. F i r s t , from column 6 of t a b l e I

( log O)max = 4.75051

( log a)min = 1.81954

Theref ore from equations (9b)

A = 1.36474

B = -4.48319

and by means of equation (sa) the were computed and a r e given i n column 8.

For i l l u s t r a t i v e purposes th ree values of Ta were chosen, 500°, 600°, and 700° F. Q j ( X i ) were computed by means of equations (Z), (lo), and ( l O a 3 , and t h e values of aj, b j , and c j were computed by equations (11). The r e s u l t s are tabu- l a t e d f o r t h i r d degree polynomial.

For each of t hese values of Ta, values of T i , CL., p j , and

Ta = 600° i n columns 9 to 1 2 of t a b l e I and i n t a b l e I1 up t o a

The values of Y a and u j were then computed by using equations ( 1 2 ) f o r each of t hese t h r e e values of Ta by f i r s t assuming m = 2, then m = 3, and f i n a l l y m = 4, corresponding to polynomials of second, t h i r d , and four th de- grees , respect ively. For each of these cases t h e standard deviat ion D w a s computed from equation ( 7 ) with S being given by equation (6 ) and Q by equation (5). The r e s u l t s a r e summarized i n t a b l e 111. The l e a s t value of D, s ign i fy ing t h e bes t fit, is obtained f o r m = 3 and T a = 600° F. The corresponding value o f . ya i s 9.5. These values, of course, correspond to equa- t i o n (13), from which t h e da ta were generated.

Application to D a t a from German Cooperative Long-Time Creep Program

As p a r t of t h e German cooperative long-time creep program, a s u f f i c i e n t amount of material of each of t h ree s t e e l s w a s supplied t o NASA t o permit t h e running of short-t ime t e s t s necessary t o pred ic t t h e r e s u l t s at long t i m e s ob- ta ined i n t h e German tes t program. t a b l e IV.

The composition of these s t e e l s i s shown i n

The resul ts of t h e NASA tests, which w e r e used i n t h e subsequent ana lys i s ,

8

f

h

Parameter, P = (T - xw))/(log t - 16.54)

Figure 1. - Master cu rve for steel K (27b KK), calculated f rom NASA data between 10 and 3700 hours.

are shown i n t a b l e V. Table V I shows t h e results of t h e long-time German tes t program. The th ree steels w i l l be designated b r i e f l y as steel K, s teel C y and s t e e l P.

With t h e use of t h e t es t da ta shown i n t a b l e V a complete ana lys i s w a s made by t h e previously described method. The general parameter discussed i n t h e INTRODUCTION w a s used, and t h e bes t values were obtained f o r t h e parametric constants f o r each of t h e three s t ee l s .

Al l t h e da t a obtained f o r these steels are shown i n t a b l e s V and V I . Many of t h e da t a poin ts were obtained f o r purposes other than t h e appl icat ion to time-temperature parameters, as described i n t h i s report . As already discussed i n references 4 and 11, a much smaller amount of da ta i s needed when an accelerated program i s desired; however, s ince these da ta were already avai lable , all the da ta indicated i n t a b l e s V and V I were used t o obtain t h e bes t possible parametric constants.

For a l l th ree steels t h e ana lys i s showed t h e s t r e s s exponent q to be zero, but t h e temperature exponent r t o be d i f fe ren t f o r each of t he three materials. For s t e e l K t he bes t value of r w a s 1, which indicated t h a t t h e bes t f i t i s obtained by t h e l i n e a r parameter. For steel P a value of r of -1 w a s obtained, which indicated a parameter s i m i l a r to t h e Larson-Miller param- e t e r ; however, t h e corresponding value of Ta w a s 200' F r a t h e r than -460° F used i n t h e Larson-Miller parameter. For s t e e l C t h e value of R w a s 2.5.

Figure 1 shows t h e results f o r steel K. Here t h e master curve cons is t s of a p l o t of s t r e s s against t he optimized parameter (T - 300)/(log t - 16.54).

Figure 2 shows t h e isothermals computed by using t h e optimized parameters, as shown on each of t n e f igures . The range of t h e NASA da ta used t o obtain these parameters i s also shown on each of t h e figures. The da ta points shown a r e t h e German results obtained to date. Tne predict ions up to 100 000 hours from t h e NASA data based on t h e optimized parameters agree wel l with t h e German data, if s c a t t e r and differences i n t e s t i n g technique between t h e two organizations are considered.

Figure 3 shows a comparison f o r each of t h e t h r e e steels between t h e bes t l i n e a r parameter, t h e bes t Larson-Miller parameter, and t h e bes t general param- e t e r . Although f o r some of t h e steels fa i r agreement can be obtained with one or t h e o ther of these parameters, it i s c l ea r t h a t t h e general parameter is superior when d l t h e materials are considered jo in t ly . If any one of t h e spec ia l cases of this parameter i s to be chosen f o r a l l materials, t h e l i n e a r

9

\ \

Predicted - German data above 5 hr

100 102 104 11 0

(a) Steel K (27b KK); parameter, P = (log t - 16.54)/(T - 300).

100 102 1

\

100 102 1

i T(No fai l ,

?6 4F-- fai lure

106 ~.

Time, h r

(b) Steel C (23b CK); param ter, (c) Steel P (14a PA); parameter, P = (log t - 8.87)/(T + P = (T - 2OO)(log t + 11.13).

F igure 2. - Analysis of German steel data by general ized parameter w i t h o p t i m u T constants (where T is temperature, and t is t i m e to rupture).

LV

(a) Steel K (27b KK).

40

20

0 100 102 104 106

Time, hr

(b) Steel C (23b CK).

Temperature f o r German data above 5 hr,

"F

0 932 0 1022 0 1112 A 1202 0 1292

(c) Steel P (14a PA).

Figure 3. - Ana lys is of German steel data by several parameters (where T i s temperature, and t i s t ime to rup ture) .

10

I I

10 -1 l l Strain,

percent

.- v) CL

0- 10 m- v)

E c m

lfl

0 -

A

1 2 5

\ h J ~~ _ _

2200 3000 100 150 200 250 300 350 400 Temperature, T, OF T - 1400 Parameter, P = - log - ~ t - 5.76

(a) 5-Percent strain. (b) Master cu rves obtained for I-, 2-, and 5-percent strain.

Figure 4. -Ana lys i s of creep data for columbium al loy FS-85 by l i nea r parameter.

parameter would appear t o be t h e bes t choice.

Application t o Creep D a t a

Although t h e r e i s no fundamental reason why t h e same parameter i s capable of representing both creep and rupture da ta , it has nevertheless been found empir ical ly ( r e f s . 1 and 2 ) t h a t t h e dual r o l e of t h e same parameter leads t o reasonable r e su l t s . Experimental data f o r creep a r e much more l imi ted , however, than t h a t f o r rupture , and such da ta tend t o contain more s c a t t e r ; hence, ana lys i s of creep da ta by t h e parametric approach has been l imi ted i n t h e past .

The method of t he present repor t can be applied d i r e c t l y t o creep d a t a without any change. A l l t h a t i s necessary i s t o redef ine t as t h e time t o a t t a i n a spec i f ied amount of creep r a t h e r than as t h e rupture time. i s assumed t h a t f o r a given amount of creep, say 1 percent , a p l o t of log CT

agains t a parameter, such as t h a t given by equation (1) , w i l l produce a s i n g l e master curve. For a d i f f e r e n t amount of creep, say 5 percent, a d i f f e r e n t master curve can be obtained, but it i s assumed t h a t t h e parametric constants, such as log t a t a ined from rupture data.

Thus, it

and Ta, remain t h e same and t h a t they equal t h e values ob-

Calculations of t h i s type were performed for columbium a l loy FS-85. The creep t e s t s were l imi t ed t o runs of approximately 1000 hours; t h e da ta a r e

11

given i n t a b l e 'm1, as taken from reference 8. Figure 4(a) shows t h e data f o r 5-percent creep s t r a i n , and f i g u r e 4(b) shows t h e master curves obtaFned for 1-, 2-> and 5-percent s t r a i n as well as t h e parametric constants obtained by t h e method of t h i s repor t . While s c a t t e r i n t h e creep d a t a i s high, t he cor- r e l a t i o n must be regarded as good. I n general , t h e poin ts agree well with t h e master curve.

Although these r e s u l t s are encouraging, much more work i s necessary before it can be concluded t h a t t h e parametric approach i s completely va l id f o r creep data. If it i s eventual ly concluded t h a t t h e parametric approach i s v a l i d f o r creep da ta and i n p a r t i c u l a r t h a t t he parametric constants a r e t h e same f o r both the creep and rupture processes, it i s obvious t h a t a g rea t saving i n t e s t f a c i l i t i e s and t e s t program planning w i l l r e s u l t . It the re fo re seems very worthwhile i n f u t u r e s tud ie s t o give more a t t e n t i o n t o t h e cor re la t ion and extrapolat ion of creep da ta by the parametric method.

Lewis Research Center, National Aeronautics and Space Administration,

Cleveland, Ohio, May 3, 1935.

12

APPENDIX A

ORTHOGONAL POLYNOMIALS AND LEAST-SQUARES DEFERNDI"I0N

O F PAl3AMETFiIC CONSTANTS

A se t of polynomials Qj(x) are s a i d t o be orthogonal over an i n t e r v a l with respec t t o t h e weighting funct ion r e l a t i o n

z(X) i f they s a t i s f y t h e following

Simi la r ly a s e t of polynomials can be defined t o be orthogonal over a set of n d i s c r e t e poin ts xi by t h e following r e l a t i o n

It can be shown (ref . 6 ) , t h a t all orthogonal polynomials s a t i s f y a three-term recurrence r e l a t i o n of t h e form

Thus by s t a r t i n g with Ql = 1 and p1 = 0 an i n f i n i t e set of orthogonal polynomials can be generated by means of equation ( A 3 ) i f values f o r % and Pk are known. (Al) o r (AZ)). From the r e l a t i o n ( A 2 ) it follows that

These can be determined from t h e or thogonal i ty conditions (eqs.

i=l

and

When t h e recurrence r e l a t i o n (A3) i s used t o e l iminate Qk+l, t h e r e i s obtained

n

i=l

13

When t h e or thogonal i ty condition (A2) i s used, equations (&a) and (A5b) reduce to

i=l

Solving equations (A6) f o r and pk gives

i=l n a k =

i=l

n

XiT?QkQk-l id

'k = n

i=l

Thus a s e t of orthogonal polynomials can be generated t h a t are orthogonal over a f i n i t e set of d i s c r e t e values of t h e variable x. Note t h a t these values need not be equal ly spaced, a condition t h a t is obviously necessary f o r stress- rupture data.

Scal ing of Polynomial Argument

From t h e recurrence r e l a t i o n (A3) with Q1 = 1, it follows t h a t t h e lead- i ng term of Qk+l(xi) i s xi. k Therefore, depending on t h e values of xi, t h e

values of This procedure can l ead to a loss of s ign i f i can t f i gu res i n performing t h e calculat ions. It i s shown i n reference 7, by comparison with t h e Chebyshov polynomials, t h a t if x i s sca led so t h a t a l l t h e values of l i e between 2 and -2, t h e polynomial

Qk+l(xi) can become very l a r g e o r very s m a l l .

14

values scal ing, $e t xmax be t h e m a x i m u m value of log a value of log a; then l e t

Q . ( q ) w i l l a l l be of approximately uniform s ize . To perform t h i s and %in be t h e minimum

-2 = kn + B

and solving for A and B r e s u l t s i n equations (9b).

It has been found i n p rac t i ce t h a t scal ing the values of x as indicated does indeed preserve t h e s ign i f icance of t h e calculat ions.

Least-Squares Procedure

I n terms of t he orthogonal polynomials, equation (3) can be wr i t ten

m + l Y = "qYa + 7 ujQj(X)

j=1

To f i n d t h e bes t values of ya and u j t h a t f i t t h e data, t h e sum of t h e squares of t h e res idua ls i s minimized. Thus l e t

Then i n order t o f ind t h e values of ya and uj t h a t will make S a minimum, S i s d i f f e ren t i a t ed i n t u r n with respect t o ya and each u and t h e r e su l t i ng equations a re s e t equal t o zero. When t h i s i s done, t h e following s e t of equa- t i o n s i s obtained:

jy

7 % Y a + a l U 1 + a2U2 + * * - + %+lUm+l = CO

?ya + blul + 0 + . . . + 0 = c1

c2 a2ya + 0 + b2u2 + . . . + 0 =

where

15

n

= cr%.Q.(Xi) j = 1 , 2 . . . m + l “ j I - l J

i=l

n b . J = T B Q ~ ( ~ ) j = 1 , 2 . . . m + l

i=l

i=l

n c = ziyiQj(Xi) j = 1 , 2 . . . m + l j

i=l

It is to be noted t h a t t h e only nonzero elements i n t h e coef f ic ien t matrix of equations (A12) a r e t h e diagonal elements and t h e elements of t h e f irst row and f i r s t column. All t h e o the r elements are zero because of t h e or thogonal i ty proper t ies of t h e polynomials used. This i s one of t h e major advantages i n using orthogonal polynomials. ments of t h e f i rs t row and f irst column, except f o r t h e f irst element, would a l s o be zero; and t h e equations would be completely uncoupled, each being computed completely independent of t h e o thers , without t h e necess i ty of solving any sets of equations with t h e r e s u l t a n t l o s s of s ign i f i can t f igures . I n t h i s p a r t i c u l a r case because of t h e added constant Y a , t h e equations are not completely uncoupled, bu t they a r e very near ly uncoupled and can r ead i ly be solved.

I n t h e usual case of data f i t t i n g , a l l t h e ele-

u j

Thus for any equation af ter t h e f i r s t

Subs t i tu t ing i n t o t h e f i rs t equation and solving for Y a give immediately

16

APPENDIX B

FORTRAN N PROGRAM

B I D Y A G 1 2 0 2 E g N t S T KOL3CRTS9 J K . - 1 4 0 M - S - P A X 6 1 5 2 B L I B S I O CON I I N U C b I E J O t l S U U R C E d I6 F T C P R M T R 1 L 1 S T 9 K E F t O t C K C C C c. C C C C c C C C c C C C c c C C C c L C C C C C c c C C i C c C C C C C C C L c C C C C C C C C C C C c C C

L U E E P / > T R E S S - K U P T U R E P A R A M E T E K P K O G K A M P R K l P K M T

P K P T DO S T A N D A R D L , € V I A T I I - l N P U k T K K D E L R E F O F F R E E O U Y P R M T K M NUMt3ER OF V A L U E S O F M R E A L , P K P T K U N U M B E R OF V A L U E S O F I) R t A O P R C T K K FuUPlHCK UF V A L U E S O F H R E n U P K P T K T A N U M B E R O F V A L U t S O F T T A K L A O P R H T M U E t i R E E P U L Y N U P I A L P K P l IV N U M B E R ( I F D A l A P O I N T 5 P K M T PP P A K A M E T E Y P R P T d S T R E S S E X P U N t N T PKMT Od P O L Y Y U V I A L P K K T n T t M P E R A T U K F t X P O N E N T P R N T R A T I U A B b ( Y - Y Y ) / D U P K K T 5 I LIMA 5 T R t S S P K N T 5 I t i d S l u P A * * O PI<MT T 1 I M t P K M T T A T I M E I N T t f l C E P T P K P T T A U S I b M A * * O * I T T - T T A ) * * K P K M T T A U S d R T A U * * 2 PRCIT T I M k C A L C i J L A T E d T ( l O . * * Y Y ) P K M T

T T A P R M T X LUb S I G N A P K M T Y Lob T P U k T Y A L U b T A PKMT Y Y C A L C U L A T E 3 L U G T P R M T

P R K I A L L L U A & T I T I E h I h C U M M U Q W I T 1 4 THI5 PKUGRAlV, A"\L) T l l l S P A P t l i P H M l A ? : P t P R k S F N T t O B Y T : i E SAME 5YMt3C)L, W I T H K F P t A T t t ~ P R M T

L t T T t R h I N D I L A T I ' \ I G lrll- U P P t K C A S t A N U I ; % t t K L t l I L H 5 4 E l V U \ P L L L L ' l - P A i < T O U T . P R M T

P K K T P K k T

P K M T S E L E C T > P A R A M E T E R P R D D U C I N L S M A L L E S T R t S I O U A L A h d GUTPUT:, A P R M l C O M P L E T E T A B L E . K F S U L T S O F A L L U T H t R V A L U E S A R F S U M K A R I Z E D I'd P K F i T A S H O A T E K T A t r L E . P K M T

P K M T * ~ + * ~ + + ~ * + a + ~ ~ ~ ~ ~ f f * n C n r c a + , r + a P R M T

P R M T T l T L t L A K D ~ VIOL)€ C A R D , A N D F I V E ( 5 ) S E T S O F D A T l i . A T T i l e [NU UF P R K T E A C H S t T O F u A T A M J L T d E A C A R U W I T H T t i t WURO ' t U U ' I N T H E F I R S T P R P T

M U S T H A V E t l L A N K S I N T H E F I R S T T H R E E C O L U M N S . L U L U h , \ h 7 3 - 8 0 A R E P R M T 1 N L O A E I ) . P R K T

P R M l T I T L E - A N Y A L P H A M E R I C I i X F ! 3 K M A T I U N - - H E A D 5 t A C t i P A G E O F U U I P U T P R M T

P K k T

N O M E N C L A T U R E I S A S F D L L O h ' S P u K T

Tr T E M P E R A T U K E P n N i T Ef iP E R A T U R E I ii T t R C t I' T

V U O G K A M E X T R A P O L A T E S CR E E P / S T K t S S - A U P 1 UR L .)A I A U 5 1 ' U C A

P P = ( Y / S 1 GMA * ' 0 - Y A / ( T T- 1 T A 1 * * K , G E I L E R A L I L E U P A K A M t T t K P r < r I

IHqtE COLUMN>. A L L D A T A C A ~ D S ( E X C F P T I Y L TITLE ANI) vni)E C A R D S ) P R V T

M O D E C A R D - b N E D F I t i R t i k WORDS I N C U L U M N 5 1 - 6 9 ' L A K S J N ' , ' L I N € A R ' p P R M T OR ' G E N R A L ' . T t i I s C A R D DEFIYES ' K K ' , T ~ I E D E G R E L OF P K w i E U E E D O M , U S E D I N C A L C U L A T I N G GUODNESS UF F I T . P f i M T

P R K I U A T A S k T 1 - - V A L U E S U F T r A T I 1 t3E I N V t S T I G A T k D - - D N F P t K C A 4 D P R K T

17

L c C C c C C L C C C c c C C c C C C C C C c c i; C c c C C C C C C C C C c C C c C C C c C C C

C

C

c

C C C c

1

F O 2 M h l ( 3 X , F l O . O 1 - - 5 0 V A L I J E S M A X I M I J M P R R T > Y P R M T 60

J A T A S t T Z - - V A L U E S UF T t M P F E A T U K E E X P O . \ l t Q T , 2, TJ J E I X V E b T I C A T F O P K M T ti UNE P E A C A K I I - - F O K M A I ( 3 X 9 F l O . O 1 - - 2 C V A L U E S H A X I M U M P R M T 62

P R K T 6 3

O N E P E R C A K D - - F O R M A T ( 3 X 9 F l O . O 1 - - 2 C V A L U E S f " lAXIPIUM P R P T 6 5 P K M T 66

Q A T A S t T 3 - - V A L J E S UF S T K F S S C X P U N E N T . ' i , T O HE 1 ' 4 V E b T I G A T E D P R P T b 4

O A T A b t T 4 - - U E G K E t 5 O F P U L Y N U H I A L , M, T O 13E I N V E S I I Z A T t d P K M T b 7 UNE P E ; ~ C A R D - - F U R M A T (~X , IZ ) - -MAXIMUY V A L U E ~ O I r o P R M T t~ E x C E t u L o - - L t R u M A Y .mr ~ J E USED. P R M T 5 9

P K M T 7 0 U A r A j t T 5 - - L ) A T A P O l U T 5 I N THE O K D E K T t M P t K A T U K t t S T I I E S S , A N 3 P R P T 7 1

1 I M E - - - c ) 1 u t > € r P L R C A K D - - F U R M A T 1 3 X t 3 F l G . 3 ) P R M T i 2 T H t V A L O E U G S T Y t S S I S A V T O M A T I C A L L Y D I V I D E 0 r3Y 103C P R M T 73 F O R A L L C A L C d L A T I U N S E X C t P l F I N U I N G THE L O G S T R L j S . P R M T 14 L O O 5 E T a M A X I M I J M . P K M T 75

P R N T 7 6 n * n a r n + a a a t r & t ~ F A T a x + a a c c + + c a r ? & M T 7 7

P R M T 7d L h C H UF T H t F I V E S t l S U F iDATA M U S T trE F U L L U W E D 6 Y A C A R U H A V I N G P A N T 79 T I i t dud(; E k D IY T t 4 E F I R b T T h K E E C O L J M N Z . P K M T PO A L L ~ A T A cAg,Js ( E x c t P r I x b r I T L L AND MOO^ C A ~ D S ) Y U ~ T H A V E i r l ~ P R M T 0 1

P K M T 8 3 ' d I T I 1 I N E A C h b E T , I I A T A M A Y -)E I N A N Y b R . I E t 7 . 1 1 H I L L 3E P X U C L S S C O 2 K C T 6 4 I:\ T l i E TJKDEg P K E S E I L T E O T i ) THE M A C H I N E . I't<E"T b 5

P R K T L O T H t L A L C U L A T I O Y S A t l t P t R F O R M t O 1 U F U U R ( 4 ) L O O P b . P K C . 1 8 7 G U I N L ; F R O M I i U Y F S M O S I T [ J L I U T F K M L I S T , Lit L i J A N T I T I t S A & E V A ? I E 0 P R M T t H It\ T H t F U L L I J r l I N G U R J E R PREAT b 9

L J E G K E ~ P C ) L Y N U I Y I A L t M P R M T 4(1 V A L U E LJF T T A P R M T 3i I E M P L R A T U K E E x P U U t 8 r l T , c( P K M T 9 2 > T d E S S t X P O N t m V T t LI P R M T ' i 3

P R M T 7 4 T H E U U r P U T T A B L E S U T I L I L t L F S S T!4A8\l 120 C 3 L U M N S 3'4 T r i E P K I k T E R P K M T 9 > Aha E X P E C T :.ILI C A 3 R I A b E C U N T R O L S O T H E K r H A l V 1, 0 , + A V O d L A N K . PRMT 96 A L I f \ l E C U J I V T t R 1 5 I I V C O R I ' U R A T E U TU L I M I T I J L I T P d T 1 3 69 L h t S P t R P K V l 3 7 P A L E . F O K E A L H U F d P A C E I H E T I T L E A N D A P P K U P R I A T E C G L U Y Q H E A ' I I U G S P K M T 3 r :

P H M T 101; P A G t C u U N T I N b ANL) E : t K O K T Y A P S M U S T i l E P R O V 1 T ) E I ) t l Y T t l E O P E K A l I N b P R E T 1 L l

P K M T l b 3 PKlJG17AoY N I T t i I d S Y S AYD I O C > P l d I L L K U N U N A 1 6 K M P C I I I U E P K M T 1 3 4

P R M T I O 5 P R M T 1 V b

L O G I C A L T R b G i C l , T 9 G G A 2 , I R Z G 3 3 P R M T 1 u 7 P K M T lC'8

d I V t 1 \ b l C & T I l L E ( 1 Z ) , T A ~ L t I S , 1 1 O ) ~ I T t r L € l 6 ~ 1 1 0 ~ P H M T I33 P R M T I I C

E U U I V A L E N C t ~ l A ~ L € ( l t 1 ) , I T ~ L t ( l ~ 1 ) ) ?REIT 1 1 L P K M T 1 1 2

COMMON / D A T A / S I G M A l L O 1 I , T I 2 0 1 ) , T T ( 201 1 P K M l 1 1 3 I / T R Y S / E A l 2 1 ~ ~ ~ ~ 5 1 ~ ~ K ~ ~ l ~ ~ T T A l 5 I I P K P T 1 1 4

3 / C A L C / P P I ~ 0 0 ) , 9 A T 1 0 ( 2 0 0 ~ ~ T I M t ~ 2 ~ ~ ) t Y Y ( Z C O ~ P Y M T 116 4 P K P T 1 1 7 5 / P L Y h P 1 L / L l T , l t N l ( 4 7 2 1 1 t Y A , I I T t l E R Z ( 6 3 ) P R M T l i b

PRPIT 1 I - J P R M T 121..

I N P U T P R M T 1 2 1 P K M T 1.2~

d.r l<ITt (6,99941 P R M l L L 3 X E A i ) ( 5 9 Y O C l ) ( T I T L t I K ) t K = l t l 2 1 P R M T 1 2 4

F I d S T T H R E E C i j L U ' I N S d L A i \ i K . P n f i r 8 c

A & t P R I N T E I ) . I ' K U G K A M € V I I S r l I 1 1 l A T R A L S F t Y TU T h L I Y I T I A L d t A 3 . P R M T

i Y i T E M . P K M l 1 2 2

2 / F D A T A / S I G : 3 ( L O O , T A U ( 200 1 , T A U b O d ( 2OC I , X ( L O P I t X X I 2.20 1 t Y ( Z O C ) PRM1 1 1 5

/ t N U / L Q*l/ Y / N / UI)/ d 3 / IO€ GK t t / D E G d E C

--

18

10

15

20

25

30

100 c C c

110 c C c C c C c C

112

K E A U ( 5 , 9 0 0 1 ) D E G R E t K = O K = K + L K t A U (> ,90021 C H t C K , T T A ( K )

I F ( C H E C K . Y f . t N D ) GO T O 10 K I A = K - 1 K = O r( = K + L R E h U ( 5 , 9 0 0 2 ) C H E L K , K ( K )

1 F ( C H t C K . N F . t N D ) GO 10 15 K K = K - 1 K = O K = K + L R E A D ( 5 , 9 0 0 2 1 C H E C K , d ( K )

I F ( C H L C K . N E . L N D 1 GCI TL) 2 0 K U -L. I(-1 K = O K = K + l A E A U ( 5 , 9 0 0 3 1 C I I F C K , M ( K I

i F ( C H L C K . N F . ~ N J ) GO T I 1 2 5 K M = K - 1 U = O K = U + l R t A 0 ( 5 , 9 0 0 4 ) C t I F C K , T T ( K ) , S I L M A ( K l r T ( K 1

I F ( L H t C K . N t . E N U ) L O TO 3 9 N - K - 1

t N 0 U F I N I ' U 1

t l N D L b G S T K t b 5 A"4U L O G T I M t

d U 100 K = l , N X ( K I = A L U L 1 0 ( z I G M A ( K ) 1 + 3 . Y ( K l = A L 0 6 1 0 ( 1 ( K ) )

L U N T INclE.

I N I T I A L I ZE C O , \ I 5 T A N l S

JUl=l. t 5 L I \ t S = ' j l I I< ti tili 3 = . F A L 5 t . IN T i l Y = 0

h C A L E L O G S OF S T K k S S

C A L L S L A L E

t-I , \d H I G H E S T d t G N t E P U L Y k O M I A L

M A X = U UO 110 K = l , K M

M A X = M A X O ( M A X , M ( K l ) L O N T I UUE

M A J O K L U U P - C A L C U L A T E S A L L Y ( A ) b A N U Y t b I L ) I J A L i w R I T t S S U M M A R Y T A D L t i - IN l )S S M A L L E S T R E b I i I U A L

UO 500 K S = l t r ( i )

C A L C U L b T F SILMA**O

19

1,

C c c

115

1 1 8 119 120

C c C

c c c

d 0 3 0 U K 3 = l r K T A

L A L C U L A T E T A U ANLJ T A U * * , ?

iJ0 1 2 0 r<=l , i \ l ILJIFF=aBS(TT(Kl-TTAiK3)1

T A U ( K I =O. b0 TCI 119

T A U ( K I = S I G O ( K l * T I ) lFF*ad( K 4 1 T a U S P H i K l = T A U ( K l * * 2

I F ( T ! J I F F l l l b r 115 ,118

L O N T I l v u E

t V A L U A T E P 1.1 L Y d 0 M I A L S

C A L L P d L Y ( P A X I

u0 200 K 2 = l . r ( M

L A L C U L A T E T i i t U K E T I C A L L O G T I r 4 l t h A X 0 T I M E S

L

C L C M P b T t K E S I O u A L L

C A L L K t S I D ( M I K 2 1 1 c L M A K E Us\€ t Q l K Y I N SUtbllVARY T A B L t L

C L i C L c

170

175

180

N T d Y = N I K Y + 1

T A d L E ( L 1 N T R Y ) = < i K 4 I

T a l l L t ( 4 , N T K Y I = T T A ( K 3 1 I A d L t i 5 , N T K Y ) = Y n T A I ~ L ~ ( ~ , N T K Y ) = J I J I R G L K L = N T R Y . E O . 2 * L LdES

r A t l L E i L, N T K Y I = L A ( K 5 I

1 T tdLb ( 3 9 N T K Y = M i K 2 )

I F ( l K b G K 2 1 bd TLI 170 bG T U 199

d L T P U T h 0 U t P A G E 13F SUMMA<Y T A d L E

u b T P b T T I T L E A N D H E A Q I Y G S F O R S U M V A R Y T A L L E

P R P l 1 % 1 P R M T 1 j L

P 2 k T L 94 P K K T 195 P Y M T l‘lt P R M T 1 3 7 P R M T 178 P K H T 199 PRMT 2 0 5 P R M T 2 b i P R M T 2 L Z P R M T 2 2 3 P K M T 2 6 4 P K M T 2 U 5 P R M T 2 2 6 P K P T 2 0 7 P R M T 2 C 8 P R M I 2 0 0 P R M T 2 1 0 P K M T 211 P K N T Z L L

PKPT 2 1 4 P K M T 2 1 5 P d M T 216 P R K T 2 1 7 P R K T I l k P K P T 2 1 9 PP.MT 2 ~ 0 P R M T 2 2 1 P K K I Z C L P R H T 2 d 3 P R M T 2 ~ 4 P K M T 225 P K P T 2 2 6 P A C T 2 ~ 7 PgP’T 2 ~ 8 P R P T L d . 7 P K P T 2 3 2 P K M T 2 3 1 PRF.!T 2 j L P 4 k T 2 3 3 P K Y , T 2 3 4 P H M T 2 3 > P R P T 2 3 b

f’Xl’JT 1 ’93

P w r z r 3

P R P T 237 P H # T 2 3 c PRP’T 23Y t‘<P,l 2 4 5 P K h T ?41 P R M T 2 4 2 P K M T 2 4 3 P R M 1 2 4 4 P R N T 2 4 5 P R M T 246 P R k T 2 4 7 P K M T 2 4 6 P R M T c 4 9 P R P T 2 7 0 P Y > l l 271

PRlVT 2 2 3 P k M T i 5 4 P R M T 2 5 5 P K P T 2 5 6

P R M T j b i

20

I

C C c

190

200 500 400 500

C C L C C

L O O 0

I F ( T R b C 4 3 ) GU T O 1000

S A V E V A L U t S P R O D U C I N G S M A L L E S T R t S I d U A L

1F ( D D i . L t . 0 0 ) GU T U 2 0 0 M l = M ( K 2 ) T T A l = T T A ( K 3 ) R 1 = K ( K 4 ) U A = U I K 5 1 r A A = YA uo1 = D O

i C N T I k U E L O N T I h d t LUNT I N u E C O h T I N b E

I F ( N T K Y . N E . 0 1 GU TU 1 7 0

E N D M A J O R L O O P

LII ITPUT O P T I M U M V A L U E S A N U H L A U l N G F O K F U L L T A d L E

T R G G R 3 = . T R U t .

L O ~ T I N U t

C C C

L C L A L C U L A T E T H t J K E I I C A L T I K E S , R A T I O 5 UF D I ~ F L ~ E U C F J i T O R k S I D U A L , A Q l I V A L U L S UF T H E P A K U P l t T E K , F U X THF C P A K A M t l E R P K U d U C I N G T H t M I N I i Y U M R t 5 l U U A L c

UC 1 0 3 5 K = l t i \ i T O I F F = n B ~ ( T T ( K ) - T T A l ) b I L U ( K ) = S I b M A ( K ) **ti1 .

I F ( T U I F F ) 1 0 3 2 , 1 0 3 1 , 1 0 3 2 1 0 3 1 l A b ( K ) = O .

1 0 3 2 1 0 3 4 T A U L L K ( K ) = I A U ( K l * * 2

b0 T O 1 0 3 4 T A U ( K ) = 5 I G & ( K ) *TJ I F F + * K 1

1 0 3 5 L O N T IKJF uu=c31 C A L L P u L Y ( P A ) L A L L Y b U b A ( M 1 ) C A L L Y l h ( M I ) C A L L R A T I O 1 C A L L P A R A M

C C U U T P U T FULL T A B L E C

K = O

A R I T E (6,9012) T T ( K ) , S I G M A ( K ) , X ( K ) , T ( K I , T l ~ I ~ ( K ) , Y ( K ) ~ Y Y ( ~ ) ~

L I N E S = L I k E S t 1

1040 K = K + l

1 R A T I U ( K ) , P P ( K )

1F ( K . E i 2 . V ) L U T O I I F ( L I i k F S . L T . 6 0 1 GU T U 1 0 4 0

w K I T k (6,9005) ( T I T L E ( K K K ) , K K K = L , l Z ) , D ~ L ~ ~ t H R I T E (6,90111 L I N E S = e ,

bC T O 1040

t k D UF P R O L R A V

P R M T 2 > 7 P K M T 2 5 8 PKk8T 0 9 P K K 1 7 (, 0 P k M T 201 P K k l Z C J ~ P d M T 2 6 5 P R k T 2b4 P R k T 2 6 5 P R M T 166 P R K I 267 P K P T 2 6 8 P R E T io') P k V l 2 1 0 P P M T Z I l P R M T 2 I L P R M T ? 1 > P R M T 2 7 4 P K h l - 2 1 5 P K P T 27.5 P K W T 2 7 7 P K P T 2 7 h P K PI T 2 I ' I P P N T i t 0 P H M J Z t , A P K P l Z U L

P K b T ? b 3 P P R T 21-14 P R h I 2 8 5 P K M T l t ' b I P R V T 2 b l PKI.:T St35 P R P T Zbir P R M T Z , > O P R M T 2 9 1 PRPrT 2 Y L I PRE 1 2 ' 1 3 P R K T 2 4 4 P K h T T35 P R K T r"i6 PRPcl 2'1 I P K M T Z ' f P P R b T c 9'1 ? K H T 3 9 L P R M T I i 1 PRP!T 3 i L P K N T 3 5 3 P R N T 3 ~ 4 P R K T !u5 P'KMT jL'6 P R r T 3 L 7 P R K I j u 8 P R M T 1 3 Y P R M I ' 5 A G P R K T 3 1 1 PHP.1 3 l L P R M T >A3 P R M T 3 1 4 P K P T 3 1 5 ? R k T 3 1 6 PRWT 3 1 I P R K T 318 P R K T 3 1 ' 9 P R M l 3 L O P R h T 3 2 1 P K K I -czz

21

c P K k T 3 ~ 3 C F O R M A T S T A T t M E N T 5 F O R P R O G R A M P R K T 3 r 4 c P R F ; l 3 2 5 c F O R M A T b F O R i i \ r P U T P R M T 3 ~ 6 C P R M l 3 ~ 7

9001 F O R M A T ( 1 2 A 6 J P R K T 3i8 9002 F U A M A T ( A 3 , F l O . O ) P R M T 329 9003 F O R M A T ( A 3 9 1 2 ) P K M T 3 3 Q 9004 F O R K A T ( A 3 ~ 0 P F 1 0 . 0 ~ 3 P F 1 0 . 0 , O P F 1 0 . 0 ) PRMl 3 3 1

L P R k T 3 3 1 C F O K M A T b F O i i G U T P U T P R M T 3 3 3 C P K K T 3 3 4 c T I T L E ( S K I P . , T U N t W P A G E ) P K W T 3 3 5 C P R M T 3 3 6

c P R M T 3 3 6

C P R M T 3 4 0

9005 F O K M k T ( l H 1 ~ 2 U X ~ l Z A 6 / 1 H , 3 0 X , A b , l O H P A R A C E T E K / l H 1 PKFIT 3 3 1

C A U M M A U Y Ol- I N P U T P K Y T 3 3 y

9006 F O K M A T ( 1 H , l O X , 4 5 t i L R E t P / R U P T U K E P A R k M E l F K b A K F I S I V t S T I b A T t U F d K / P K r T 3 4 1 1 1 H , I 2 , 1 8 H V U L U E ( S ) OF T ( A ) , , I 3 , 2 5 H T E M P t r R A T i J R t t X P t l U t N T ( S ) , t I 3 , PUP1 3 4 2 ZL4H b T R k S S E X P O U E N T ( S ) , A N O , I 3 , 1 4 H P O L Y N J M I A L ( S ) / l I l t l : ! ) i t S t I d S I \ i G ~ P K Y N T 3 4 5 314,12H D A l A P O I ' v T S / l I 1 ) P K Y l 3 4 4

C P R M T ? 4 5 C H E A D I N ~ S F G A L d M M A R Y T A B L t , U N t L I h E OF S U Y P A R Y T A G L t P K M T 3 4 5 c PHWT 3 4 7

9007 F O K M A T 9 2 ( 2 X , 1 I-1G t 7 X 9 1 HR 9 6 X 9 1 HM 7 5 X 9 4 H T ( A ) p 5 X 9 4 H Y ( A 1 9 4 X 9 PRF: 1- 3 4 8 1 8 H S T O . D E V . , I O X ) / l H 1 P K K T 3 4 9

9008 F O d P A T ( 1 H ~ 0 P F 5 . 2 ~ ~ 8 . 2 , 1 5 , F 9 . 0 1 F 1 0 . 2 , 1 P E 1 1 . 2 l P l l Z ) P R P T 3 5 b 9009 F U d k A T (lH+r~RX,O?F5.2,F~.21I51F9.O,F1C.ZllPEll-2) P K M T 3 5 1

C P R M T 3 5 Z C U P T I M U N V A L d t 2 P K M T 353 c P R K T 3 5 4

9010 F U K M A T L l H 1 0 X 4 4 H V A L U E S P R O D U C I N G S M A L L E S T STAILL7LXD i ) t V I U T I L J [ ' , / 3 H I ) u ' = P K M T 3 5 5 l F S . Z , 4 H , R = F 5 . 2 , 4 H r M = I 2 , 7 t i , T ( A ) = F 6 . 0 , 7 H , Y ( A ) = F Y . 3 , 1 1 i l , S T O . J F V . P R M T 3 5 6 2 = 1 P t 9 . L / l H O ) P K M T 3 3 7

C P R K T 5 5 6 C H E A D I N ~ S F O X F U L L T A H L E , ONE L I N t O F F U L L T A b L E P R P T 359 C P K W T 363

9 0 1 1 F O R K A T ( 5 H T t H P 1 4 X , O H S T ~ t S j , 3 X , 3 H L ~ ~ , 6 X , 4 H r I M E , ~ ~ X , 6 ~ i L A L ~ l ~ , j X , P R Y T . 3 6 1 1 3 H L O G , 3 X , 8 H C A L C L O G , Z X ~ ~ ~ ~ E V / S L J , 3 x 9 9 t I P A R A M E T I H / 1 t i T b X , t ; i i ( a t - 3 ) t 2 X t P R P 1 3 h L Z 6 H ~ T ~ E j S , 1 4 X , 4 H T I M t ~ 5 X ~ 4 H T I ~ E ~ 4 X ~ 4 ~ T l M ~ / l ~ ) P K M l 3 6 5

( 1H

9012 F O R M A T ( 1 H ~ ~ P F 5 . 0 ~ ~ 8 . 1 ~ F A . 3 ~ 2 F l C ~ l ~ 3 F Y . 3 ~ l F l Z ) P R C T 3 6 4 c ? R M T $ 6 5

9999 F O K M A T ( 1 H 1 ) P R C T 3 b b c P R Y 1 3 5 7

t ~ i i P U M T 3 b &

22

S I B F I C P R M d L K L I S T ~ R E F I D E C K C S E T S F I R S T P U L Y N O M I A L T O U N I T Y A T A L L S T A T I O N S A V D S T O R E S C A L P H A M E R I C C U D t M O R O S C

B L O C K U A T A C O M M O N / P L Y N M L / Q Q ~ 2 1 ~ 2 0 0 l ~ 0 T H E R S ~ 8 5 ~ / E N O / E N D / N A M E S / ~ A M E S ~ 2 ~ D A T A (~Q(l,K)~K=lr200)/20O*l./~END/3HEND/~

1 (NAMES(K),K=lr2)/12HLARSONLINEAR/ E N D

B I B F T C P A R A M L I S T I R E F I D E C K C S U t l R O U T I N E F O R E V A L U A T I N G T H E P A R A M E T E R A T E A C H P O I N T L

S U B R O U T I N E P A R A M

COMMON / F D A T A / S I G O ( Z O O ) ~ T A U ( 2 O O ~ ~ O T H l i R S ( 6 O O ~ ~ Y ( 2 O O l 1 /CALC/PP(200)~0THERl(6OO)/N/N 2 / P L Y N M L / O T H E K 2 ( 4 2 2 1 ) r Y A r O T H E R 3 ( 6 3 1

C UO 10 K = l r N

P P ( K ) = ( Y ( K ) - § I G Q ( K ) * Y A ) / T A U ( K ) 10 C O N T I N U E

R E T U R N E N D

C C

C

C

10

20

30

P R M B P R M B P R M B P R M B P R M B P R M B P R M U P R M B

P A R M P A R M P A R M P A K M P A R M P A R M P A R M P A R M P A R M P A K M P A R M P A R M P A R M

d I t 3 F T C Y T H L I S T I R ~ F I D E C K S U U R O U T I N E F U R C A L C U L A T I N G T I M E S A N D L O G T I M E S F R O M T H E P A R A M E T E R Y T H

S U B R O U T I N E Y T H ( M I

C U M M O N /CALC/OTHERS(400),TIME(200)~YY(200) 1 / F O A T A / § I G Q ( L O O ) ~ T A U ( 2 0 O ~ ~ O T H E R l ~ 8 0 0 ) 2 / P L Y N M L / U 0 ( 2 1 , 2 0 0 ) r U ( Z l ) , Y A , U T H t R 2 ( 6 3 ) 3 / N / N

U O 10 K = l , N Y Y ( K ) = 0.

M l = M + 1 L O N T I N u E

0 0 30 K = l , N 00 20 J = l , M l

Y Y ( K ) = Y Y ( K ) + Q O ( J , K ) * U ( J ) C O N T I N U E

Y Y ( K ) = T A U ( K ) , Y Y ( K ) + S I G O ( K ) ~ Y A T I M F ( K ) = 1 0 . * * Y Y ( K )

C O N T I N U E R E T U R N t N D

Y T H Y T H Y T H Y T H Y T H Y TH Y T H Y T H Y T H Y T H Y T H Y T H Y T H Y T H Y T H Y T H Y T H Y T H Y TH Y T H Y T H

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8 9 10 11 12 13

1 2 3 4 5 b 7 ti (3

1 0 11 12 1 3 14 1 5 16 1 7 18 19 20 21 2 2

23

p

6 I B F T C R A T I O 1 L I S T t R E F t D E C K C S U B R O U T I N E F U R C A L C U L A T I N G R A T I O S C OF I N D I V I D U A L R E S I D U A L S T O R O O T - M E A N - S Q U A R E R E S I D U A L C

S U B R O U T I N E R A T I O 1 r b

COMMON /FDATA/OTHERS(1000~tY(200) 1 / C A L C / O T H E R 1 ( 2 0 0 ) ~ R A T I O ~ 2 O O ~ ~ O T H E R 2 ( 2 O O ~ ~ Y Y ~ Z O O ~ 2 / N / N / U D / OD

C UO 10 K = l r N

R A T I O t K ) = A B S L Y ( K ) - Y Y ( K ) ) / O D 10 Z O N T I N U E

R E T U R N € N U

R A T O R A T O R A T O R A T O R A T U R A T O R A T 0 R A T O R A T O R A T O R A T 0 R A T O R A T O R A T O

B I B F T C R t S I D L I S T t R E F t O E C K C S U a R L l U T I N E FUR C A L C U L A T I N G R E S I D U A L K E S U C K t S U C T H t R E b I D U A L I S d A S E D O N T H E L O L O F T H t T I M E . R t S U C I r I S D E F I & E u A S THE S Q U A R E R O U T O F T H t S U M O F T H E S Q U A R E S O F R E S U C T H E I N D I V I D d A L R E S I U U A L S D I V I D E D B Y T H t D I F F E R L Y C t B E T W E t N T t i t ' 4 U M R t S U

C F R E E D O M , K K , D E P E N D S O N T H E P A R A M E T € R ( S t E M A I R B O O Y O F K E P O K T ) . R t S D L K K = 2 F U R L A R S U N - M I L L E R P A K A M t T E R R t S U C K K = 3 F U R L I N E A R P A R A M E T E R R E S U C K K = 5 F U R G E N t R A L P A R A M E T F K R E S D t R t S U

C B E R O F D A T A P O I N T S A N D T H E D E G R E E S OF k R E C D O M . T H E D E G R E E L O F R E S O

C C

C

C

10

20 30

40

DO = S i J R T ( ( Y - Y Y l * * 2 / ( N - M - K K ) l

S U B R O U T I N E K t S I D ( M )

COMMON /FDATA/OTHEKS(lOOO)rY(2OOJ 1 /CALC/OTHER1(600)rYY(200) 2 / O D / D u / N / N / O t G R E E / D E G R E E / N A M E S / F A M E S ( 2 )

1 F ( D E G R E E . E Q . F A M E S ( 2 ) ) GO T U 20 I F ( D € L R E E . E Q . F A M E S ( l ) ) GO T U 10

ti0 T O 30

L O T U 30

K K = 5

K K = 2

K K = 3

OD = 0 . D = N-M-KK

UO 40 K = l t N O D = D D + ( Y ( K ) - Y Y ( K ) J * * Z

C O h T I N U E DO = S h l R T ( D D / D )

RETURi' i t N D

R t S U R E S U R t S D K E S D R E S U R t S U K E S U R t S O R E S U R E S O R t S U R t S D

R t S O R E S O R t S O R E S U R t S D R t S U R k S U R t S L ) R t S U R t S D

nESu

1 2 3 4 5 6 7 8 9

1 0 11 12 13 14

1 2 3 4 5 b 7 G Y

16 1 1 1 2 13 1 4 15 Ab 1 7 18 19 iL 2 1 22 2 3 14 2 5 L b 17 LE; ZY 3C 31 3 2 3 3 3 4

24

E I B F T C Y b U d A L I S T , K t F , U t : C K c SUt3RbJ I I N E FUR E V A L d A T l N b Y ( A 1 Y3UI3 I

YSUt l L Y 5 U t j 3 YSUR 4 Y S U t , 5 Y a U t i 6 Y S U H 7 Y S U J h Y S U U Y Y S U d I C YSUO L i YSUd 1 2 Y S U 6 1 3 Y S U d 1 4 Y b U t l L', Y S U d I b Y 3 U d 1 7 Y S U d 18 YSUd 1 7 YSUY L >

Y b U d i l Y S U t l L L Y b U d L 3 YSUd c 4 Y S U d / > Y b U 6 L:,

Y S U d 27 Y S U 8 cti YSUb L 9 Y S U i l 3 0 Y S J J 3 1 Y S U d 3.2 Y S U d 3 3 YSUO 3 4 Y3UB 55 Y S U d 36 Y S U B > I Y S U B 3 8 Y S U d 3 9 Y S U 8 4.1 Y S U B 4 1 Y S d l 3 4 L Y h U d 4 3

25

5 1 b F T C P U L Y L I S T , K E F , O t Z K C C C C C C C C

c

c

10

20

30

40 50

S U S R C I d T I N E F U R k V A L u A T I Y G O K T H O b U h G L P O L Y N O M I A L S

A L L P O L Y h O M l n L S UP T O M A X I M U M U t S l r ( t D U E G R E t A R t t V A L U A T t U A T E A C h D A T A P O I N T

TH5 F I A S T P O L Y N O M I A L I S I D E N T I C A L L Y t B U A L TU U P ; I T Y T H t S t V A L U E S A R E S T O R E D B Y A B L O C K D A T A S U B R U U T I ' i E

b U t J R U u T I N E P O L Y ( M 1

COMMON / F D A T A / O T I i E R A ( 400 ) p T A U S O R I 2 0 0 ) , U T H E R Z I 2 0 0 1 9 X X I 2 0 0 ) 1 U T H E R 3 ( 2 0 0 ) 2 / P L Y N M L / ~ Q l 2 l r 2 0 0 ) , 0 T H E K S ( 4 5 ) 1 A L P H A ( 2 O ) , ~ € T ~ ( 2 0 ) 3 / N / h

P O L Y A P O L Y i P O L Y 3 P O L Y 4 P O L Y 5 P O L Y t P O L Y 7 P O L Y 8 P c j L Y 4 P O L Y 1c. P O L Y A i P O L Y 1 2 P O L Y 13 P O L Y 1 4 P b L Y 1 5 P U L Y Ab P U L Y 1 7 P O L Y i E P U L Y i s P O L Y L i '

P U L Y 2 1 P O L Y 2 2 P U L Y L 3 P O L Y 2 4 P O L Y 2 5 P U L Y L U

P U L Y 27 P O L Y i b P U L Y 2 Y P O L Y 3r, P O L Y 3 i P t i L Y 3 2 P O L Y 3 3 P U L Y 3 4 P O L Y 3 5 P U L Y 3 6 P O L Y 5 7 P b L Y 3 8 P L I L Y 3 Y P U L Y 4 L P i l L Y 4 1 P O L Y 4 2 P U L Y 4 3 P O L Y 44 P c l L Y 4 5 P O L Y 46 P O L Y 47

26

S I B F T C S C A L € L I S T t K E F , O E C K C C C C

C

C

10

20

S U d K O U T I N E FUR S C A L I N G L O G S OF S T R E S b

i H t 5 C A L t O V A L U E S L I E I N T H E K t G I U N - 2 T O 2

5 U d K U U T I h E S L A L E

COMMON

ellti = 0. S M A L L = L e t 5

/ k D A T A / O T l i t K L ( 6 0 O ) , X ( 2 0 0 ) ~ X X ( 2 0 0 ) , O T t l t R 2 ( 2 0 ' 3 ) / N / N

UO 10 K = l , N t i I G = A M A X l ( b I G , X ( K ) ) J M A L L = A M I N L ( $ M A L L , X ( K ) 1

L O N T I NU€ A = 4 . / ( b I G - > M A L L ) U = 2 . * ( ~ I G + S M ~ L L ) / ( ~ I G - S M A L L )

00 2 0 K = l , N X X ( K 1 = A e x ( ~ 1 - B

C O N T I :QUE R E T U R N t i l 0

S C A L 1 S C A L i: S C A L 3 S C A L 4 S C A L 5 S C A L t S C A L 7 S C A L i: S C A L 9 S C A L L O S C A L 11 S C A L 1.2 S C A L 1 3 S C A L 14 S C A L 1 5

5 C A L 1 7 S C A L 18 5 C A L 19 $ C A L LC, S C A L 2 1

S C A L It,

27

IllllllllIlIIllll IIIIIIIIII I I 1 I 1 Ill1 I II lIIm~11ll1l1l 1l1ll1l II Ill

REF'EIIENCES

1. Larson, F. R.; and Miller, James: A Time-Temperature Relationship for Rupture and Creep Stress. Trans. ASME, vol. 74, no. 5, July 1952, pp. 765-771; Discussion, pp. 171-175.

2. Manson, S. S. ; and Haferd, Angela M. : A Linear Time-Temperature Relation NACA TN 2890, 1953. for Extrapolation of Creep and Stress-Rupture Data.

3. Orr, Ramond L. j Sherby, Oleg D. ; and Dorn, John E. : Correlations of Rupture Data for Metals at Elevated Temperatures. Trans. ASM, vol. 46, 1954, pp. 113-128.

4. Manson, S. S.: Design Considerations for Long Life at Elevated Tempera- tures. NASA TP 1-63, 1963.

5. Manson, S. S.; and Mendelson, A.: Optimization of Parametric Constants for Creep-Rupture Data by Means of Least Squares. NASA MEMO 3-10-593, 1959.

6. Hamming, R. W.: Numerical Methods for Scientists and Engineers. McGraw- Hill Book Co., Inc., 1962, pp. 223-246.

7. Forsythe, G. E.: Generation and Use of Orthogonal Polynomials for Data- Fitting with a Digital Computer. J. SOC. Indust. Appl. Math., vol. 5, no. 2, June 1957, pp. 74-88.

8. Titran, Robert H. ; and Hall, Robert W. : High-Temperature Creep Behavior of a Columbium Alloy, FS-85, NASA TN D-2885, 1965.

9. Murry, G.: Extrapolation of the Results of Creep Tests by Means of Para- metric Formulae. Vol. 1 of Proc. Joint Int. Conf. on Creep, Inst. Mech. Eng. , 1963, pp. 6-87 - 6-100.

10. Manson, S. S. j and Brown, W. F. , Jr. : Time-Temperature-Stress Relations for the Correlation of Extrapolation of Stress-Rupture Data. Proc. ASTM, vola 53, 1953, pp. 693-719.

11. Manson, S. S. j Succop, G. j and Brown, W. F. , Jr. : The Application of Time- Temperature Parameters to Accelerated Creep-Rupture Testing. Trans. ASM, V O ~ . 51, 1959, pp. 911-934.

28

1

Index, i

1 2 3 4 5 6 7 8

Time, t, h r

4954.68 L1365.9

2908.

1340.

625.342

117.371

34.4856 995.25

TABLE I. - CALCULATION OF POLYNOMIALS FOR THEORETICAL DA!l?A FOR IMLRD DEGREE WLYNOMlY&

[Temperature intercept, Ta, 600' F. 1

Stress , 0, p s i

56 300 1 9 800 30 300

5 080 1 2 900

778 4 190

66

2

Tempera-

T, OF

ture,

1100 1100 1200 1200 1300 1300 1400 1400

0 1 2 3 4

3 1 4

- - - - - - - 0.27092 -.57813

.28432

.41260

5

l o g t

3.69501 4.05560 2.79612 3.46359 2.06956 3.12710 1.53764 2.99793

6

log a

4.75051 4.29666 4.48144 3.70586 4.11059 2.89098 3.62221 1.81954

7

q(T-Ta)l

500 500 600 600 700 700 800 800

8

Scaled 1% 0, X

2. 0 1.3806 1.6328

.57433 1.1267 -.53777

.46017 -2.0

I 9 1 10

Polynomial

Ql

1 1 1 1 1 1 1 1

TABLE 11. - INTERMEDIATE CALCULA'LIONS FOR THEORETICAL

DATA FOR THIRD DEGREE POLYNOMIAL

[Temperature intercept, Ta, 600' F. 1

Q2

1.3619 .30341 .e5576 -. 80869 . le925

-2.2709 2.8030

.51879

l l 2 Index, I a

J I 3

P

------- 0. 1.6548 1. 3315

.a0618

4

a

8.0

786.18 465.68 286.01

5200.

5

b

- - - - - - - - 3.48 X l O t 5.7589 7.6678 6.1816

6

C

23.743 14897.

2616. 3796.9 2694.6

Q3

-0.19594 -1.6875 -1.4583 1.5741 2.5067 -. 90880

-. 78251 .015445

-.57205 1.7195

-1.0745 .92564

-. 77354 -. 98133

29

TABLE 111. - FIT FOR SEXEFLAL VALUES OF LINEAR

PARAMETER FOR THEORETICAL DATA

Degree of polynomial

Temperature,

Ta

500 500 500

600 600 600

700 700 700

Variab le ,

y a

10.54 10.54 10.55

9.49 9.50 9.50 a. 44 8.46 8.45

Devia t ion

0.008049

.010660

.004859

.000002

.000003

.015412

.013937

.014469

.0097a6

TABLE I V . - COMPOSITION OF STEELS RECEIVED

FROM GERMAN COOPERATIVE LONG-

TIME CREEP PROGRAM

[&-received, 20-mm-diam. b a r stock. 1

Element

Carbon S i l i c o n Manganese Chromium Molybdenum

Columbium and tan ta lum

Nickel Titanium Vanadium Tungsten

I Composition, pe rcen t

I C

(23b CK)

0.065 .47 .60

17.24 2.08

.02

11. 90 .39 .10

Less than 0.005

S t e e l

P (14a PA)

0.270 .26 .60 2. 62 .27

Trace

.14 Trace .26

Trace

~

K (27b KK)

0.068 .45 .73

16.14

2.10 I *44 I

13.12 Trace .05

Trace

30

TABLE V. - NASA fi- DATA

(a) S t e e l K (27b KK)

S t r e s s , 0, p s i

20 000.000 20 000.000 20 000.000 20 000.000 20 000.000

20 000.000

eo 000.000 20 000.000 20 000.000

40 000.000 40 000.000 40 000.000 40 000.000 40 000.000

40 000.000 40 000.000 40 000.000 60 000.000 60 000.000 60 090.000 60 000.000 60 000.000

20 ooo.ooo

Temperature, T, OF

1022.00 a1022. 00 alo22. 00 a1022.00 a1022.00

a1022.00 "1022.00 a1022. 00 a1022.00 a1022.00

aloaz. 00 a1022.00 1112.00 1112.00 1112.00

"1112.00 a1112.00 "1112.00 a1112.00 a1 112.00 a1112.00 a1112. 00

1112.00

Time,

hr t ,

0.400 1.900 4.450

23.700 25.500

38. 136. eo0 394.800 704.600

1212.

2.700 7.500

15.203 44.400

377,

1417. 2110. 5367.

.610 1.250 4.400 4.500

10. :>OO

lemperature T, ??

a1600. 00 a1620. 00 a1660. 00 "1680.00 a1700. 00

"1740.00 a17P0. 00 a1425. 00 a1450. 00 "1480. 00


77 000.000 72 500.000 72 000.000 70 000.000 68 000.000

66 000.000 66 000.000 65 030.000 62 500.000 60 000.000

60 000.000 55 000.000 68 000.000 65 000.000 62 500.000

60 000.000 57 500.000 55 000.000 52 500.000 50 000.000 45 000.000 43 000.000 37 000.000

a1500. 00 a1520. 00 "1560.00

T i m e ,

hr t,

1.500 13.800 10. 36. 700 60.400

73.300 107.600 201.300 250.400 990.

817.500

.750 2.250 4.300

13.900 22.700 51.500

147.500 2R3.

1 020. 1 579.

1 3 1 4 0 .

3 680.

"1570.00 a1600. 00

1650.00 1700.00

a1260. 00 a1202.00

a1290.00

"1292.00 a1292.00 a1320. 00 a1360. 00 a1400. 00

1440.00 i4eo .m

a1112.00 "1160. 00 allso. 00 a1202.00

? t r e s s , 0, p s i

5 000.000 5 000.000 5 000.000 5 000.000 5 000.000

5 000.000 5 000.000 1 000.000 1 000.000 1 000.000

1 000.000 1 000.000 1 000.000 1 030.000 1 000.000

1 000.000 1 000.000 1 000.000 1 000.000 1 000.000

1 000.000 1 000.000 1 000.000 1 000.000 1 000.000

1 000.000 1 000.000 ) 000.000 1 000.000 ) 000.000 ) 000.000

Time, t , hr

570.200

156.800 91.600 62.700

40.500 10.600

1690. 550. 300 270.

170. 120.500 40. ;71.500 15. ROO

5. 250 1.750

3307. 667.400 255.

347.100 363. 180.400

28.900

ie6.600

82.

9. 2.500

4258. 1110.

696. 300 350.

Temperature, T, OF

1600.00 1560.00 1520.00

"1480.00 ?460.00

y440 .00 ?400.00 a1360. 00 "1340.00 "1320. 00

1320.00 1290.00

&IZSO. 00 a1230. 00 a1170. 00

"1140.00 "1125.00 112.00 1200.00 1170.00 1150.00 1140.00

al l20.00

(b) S t e e l C (23b CK)

Temperature, T, OF

a1230. 00 a1250. 00 a12eo. 00

1292.00 a1310. 00

a1112. 00 a1120.00

a1202.00

a1202.00 a1210.00 a1220.00

a1150.00 a1170. 00

a1240. 00 1270.00

12bO. 00 1292.00 1300.00

a1112.00 a1112.00

~1l12.00 a1112. 00 a1112. 00 "1112.00 a1112.00

a1112.00 "1112.00 a1202.00 a1202.00 a1202.00

S t r e s s , 0, ps i

30 000.000 30 000.000 30 000.000 30 000.000 30 000.000

40 000.000 40 000.000 40 000.000 40 000.000 40 000.000

40 000.000 40 000.000 40 000.000 40 000.000 40 000.000

40 000.000 40 000.000 40 000.000 34 000.000 43 000.000

46 000.000 46 000.000

50 000.000 52 000.000

54 000.000 57 000.000 25 000.000 34 000.000 35 000.000

48 ooo.ooo

.- Time,

h r

175. 700 103.500

t,

58.100 !1 300.

22.500

667.900 785.400 266.700 127. 900

44.100

74. 40.500 37. 800 17. 200

4.500

1.200 1.300 .e00

363.100

233.900 261.400 1P3.100

65.600

39.300 23.300

199.400 124.300

2 274.

84.500

1 0 7 4 .

Temperature, T, OF

a1112.00 a1110.00 a1080.00 aioeo. 00 a1050. 00

a1030. 00 a1022.00 a1020.00

1040.00 1022.00

a1000.00 a9s0.00 a960. 03 a940. 00 a920.00

a1120.00 "1200.00 al28O. 00 "1340.00 "1500.00 a15G0. 00 "l5nO. 00 "1540.00

Temperature T, O F

a1202. 00 a1202. 00 "1202.00 a1202.00 a1202.00

1202.00 1202.00 1202.00 1202.00

a 1 2 . a 00

al2!~2.00 a1232.00 al2:~2.00 ?2.'2.00 a12.'2.00

11292.00 "1292.00

1292.00 12!'2.00 1292.00

a1060. 00 a1300. 00 a1360. 00 a1430. 00 ai4eo. 00

a1570. 00 a1630. 00 a1140. 00 a1320. 00 a1480. 00 a1540.00


$0 000.000 ;o 000.000 50 000.000 $0 000.000 ;o 000.000

50 000.000 $0 000.000 j0 000.000 15 000.000 75 000.000

75 000.000 75 000.000 75 000.000 '5 000.000 75 000.000

50 000.000 LO 000.000 50 000.000 ?5 000.000 .5 000.000 12 000.000 !O 000.000 .o 000.000


36 000.000

12 000.000 1 4 000.000 65 000.000

4G 000.000 18 000.000 19 000.000 50 000.000 10 000.000

23 000.000 25 000.000 2Q 000.000 29 000.000 32 000.000

33 000.000 34 000.000 36 000.000 37 000.000

38 ooo.ooo

38 ooo.ooo 60 000.000 25 000.000 19 000.000 15 000.000 1 2 000.000

8 000.000 6 000.000

34 000.000 15 000.000

8 000.000 6 000.000

Ti me ,

h r t,

12.900 34. 52.200 37.400

239.

445. 989.900 817.500

.330 5.850

15.600 46.500

138. 542. 579.600

186.100 130.203 132.700 125.800

51.300 41.700 32.400 148.200 -

~- Time, t , hr

68.601 59.30( 24.401 14.5M 22.901

~

7. 2. R51 2.551 1.47(

t'5!1. 701

1!>4. 601 75. 34. GO1 31. 13.30

19. ROI 10. 401

2.751 7.60' 1.65'

42.50 89.60 35. 71.40

147.90

104. 140. 90

1077. 1505. 2237. 1258. -

lata point i n parametric analysis .

31

TABLE V. - Concluded. NASA FUPITRE DATA

(c) Steel P (14” PA)

Cemperature

3 932.00

&932.00 “932.00 “932.00 “932.00

“932.00 “932.00 “932.00

932.00 1022.00

1022.00 1022.00 1022.00 1022.00 1022.00

1022.00 “1022.00 9022.00 a1022.00 1415.00 1340.00 1315.00 129o:oo

Stress,

p s i

65 000.000 60.000.000 60 000.000 51 500.000 55 000.000

52 500.000 40 000.000 30 000.000 27 000.000 58 000.000

55 000.000 50 000.000 47 000.000 47 000.030 45 000.000

42 500.000 40 000.000 37 500.000 25 000.000 10 000.000 10 000.000 10 000.000 LO 000.000

0,

-. -. ~-

Time, t, h r -

3.800 14.150 14.400 10. 18.900

51. 623.

7 592. 11 410.

.580

.717 1.280 2.450 6.200 3.500

6.300 22.500 12 .

382.200 .170

1.500 3.700 6.100

Temperature T, OF

“1250.00 “1220.00 “U80. 00

1140.00

“1090.00 1100.00 1080.00 1060.00 1050.00

1040.00

p o . 00

“lO20.00 9010.00 “1000.00 “990.00

”980.00 “960.00 “940. 00 a930. 00 %oo. 00 932.00 as7.00

a860. 00

Stress, =, ps i

10 000.00c 10 0oo.ooC 10 00o.ooc 10 000.00c 10 000.00c

10 000.000 40 000.000 40 000.000 40 000.000 40 000.000

40 000.000 40 000.000 40 000.000 40 000.000 40 000.000

40 000.000 40 000.000 40 000.000 40 000.000 40 000.000 70 000.000 70 000.000 70 000.000

-

Time,

h r

19.20c 42.

167. 203.4oC 608.

2639.

t,

1.30C 2.20c 4.30C 6.800

7.400 22.500 20.100 63.300 51.200

80.603 192.100 427.900 623.

372. 1.400 5.800

31.200

~. -

Temperature, T, OF

“740.00 &785. 00

“880. 00 a932. 00

“~20. 00

“1022. 00 “1050.00 “1090.00 “1090.00 “1120.00

“1160. 00 &1230.00 “1290.00 “740.00 “780.00

a830. 00 “880. 00 “932.00 a980. 00

”1000.00 “1030.00 1070.00 “1150. 00 “1220.00

-.

-. .~

Stress, 0, ps i

90 000.00(

70 ooO.OO( 60 000.00( 50 000.00C

30 000. OOC 25 0OO.OOC 20 0oo.ooc 20 000.00c 16 000.OOC

13 0OO.OOC 8 000. ooc 5 000.00c

30 000.000 70 000. OOC

50 000.000 50.000.000 35 000.000 30 000.000 35 000.000 !O 000.000 L6 000.000 8 000.000 5 000.000

ao ooo.oo(

Time, t, hi”

57.100

195.800 120. 103.500

186.700 123.500 79.500

112.400 183.500

100.300 97.900

139.700 996.600

84.

1122.

948.800 599.

1902. 754.800 970.700

804.800 948.500 960.

1084.

Data point used i n parametric analysis.

32

TABU VI . - GERMAN RUPTURF: DATA

remperature, S t ress , I Time, - T 4

1022.00 1022.00 1022.00 1112 .00 1112.00 1112.00 1112.00 1112.00

1112.00 1112.00 1112.00 1112.00 1112.00

1112.00 1112.00 1112.00 1112.00 1202.00

1202.00 1202.00 1202.00 1202.00 1202.00

1202.00 1202.00 1292.00 1292.00 1292.00

t, p s i I hr

Stee l K (27b KK)

76 899.999 0.100 66 899.999

72 500.000 64 000.000 55 500.000

S t e e l C (23b CK) ~

14 200.000 17 800.000 28 400.000 28 400.000 28 400.000

35 600.000 44 100.000 5 1 200.000 59 800.000 11 400.000

1 4 200.000 17 800.000 22 800.000 28 400.000 35 600.000

42 700.000 52 600.000 ll 400.000 11 400.000 13 900.000

60 000. 30 000. 3 500. 3 000. 2 200.

1 200. 520. 150.

82 790.

1 5 000. 6 500. 1 800.

550. 124.

5.

30 000. 20 000. 4 500.

.loo

. 100

Temperature,

1292.00 1292.00 1292.00 1292.00 1292.00 1292.00

-

932.00 932.00 932.00 932.00 932.00

932.00 932.00 932.00 932.00

1022.00

1022.00 1022.00 1022.00 1022.00 1022.00

1022.00 1022.00 1022.00 1022.00 1022.00 1022.00 1022.00

S t ress , 0, p s i

Time, t, hr

a t e e l C (23b CK)

17 21 800 4 0 0 : O O O ~ 000 1 100.

21 400.000 27 000.000 180. 27 000.000 140. 47 000.000 .loo ;eel P (14a PA)

84 000.000 75 500.000 7 8 399.999 55 500.000 44 100.000

34 200.000 27 000.000 22 800.000 17 100.000 72 599.999

69 699.999 65 500.000 59 800.000 35 600.000 27 000.000

22 800.000 22 800.000 17 100.000 13 900.000 13 900.000 11 100.000 8 830.000

~-

0.100 .loo

2. 150.

1 700.

2 600. 16 000. 22 000. 100 000.

.loo

.loo 1.200 1.500

150. 300.

400. 900.

2 100. 6 500. 8 000.

10 000. 68 000.

33

i

I i

Ilemperature;

~

T, OF

2005 1900 1790 2175 2400

2300 2200 2100 2100 2000

2000 2000 25 75 2200 2800

2620 2200 2 900 3000 2450

TABU3 VII. - CRFEP DATA FOR COLUMBIUM ALLOY FS-85

~


25 000 25 000 25 000

10 000

10 000 10 000 10 000 10 000 10 000

10 000 8 500 6 000 6 000 4 000

4 000 4 000 3 000 2 000 2 000

~

18 ooo

1-Percent creex

0 .6 26.

210. 4 .9 3 .4

25.4 54.

355. 380 775.

900. 2480.

425. 5 .6

3 .4

14 .4

2. 6 4. 6

1140.

------

Time, t, hr

2-Percent c reep

3.0 33.

257. 7. a 5 . 7

41. 84.

500. 570.

1325.

1420.

10. 710.

------

6. 4

5-Percent c reep

6 . 1 45.

332. 13. 10.8

68. 133. 765. 875.

2175.

------ 22.2

13.5 1370.

56.

13. 8 33. 2

------

950.

34 NASA-Langley, 1965 E-2781

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Optimization of Time-temperature Parameters

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