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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
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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 determin- ing 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 poly- nomial, 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 (appen- dix A). The use of orthogonal polynomials for representing discrete sets of unequally spaced data is described in reference 6 and in more detail in refer- ence 7. A further improvement can be obtained by performing a linear trans- formation 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 gen- eral 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 ob- tained. 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 poly- nomials 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 de- termined. 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 ad- vantages 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 cor- responding 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 pro- gram. 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 accel- erated 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 para- metric 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 poly- nomials 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
S t r e s s , 0, p s i
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
S t r e s s , 0, p s i
$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
S t r e s s , 0, p s i
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
~
S t r e s s , 0, p s i
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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