A comprehensive creep model

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A comprehensive creep model

Harmathy, T. Z.

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U K l

TH1

N21r2

no.

324 c . 2

BLDG

NATIONAL

RESEARCH CouNclL OF CANADA

CONSElL NATIONAL DE RECHERCHEs DU CANADA

A @ " ~ f @ a ~ n

A COMPREHENSIVE CREEP MODEL

by

T. Z.

Harmathy

Reprinted from Transactions of the ASME Journal of Basic Engineering Vol. 89, Series D, No. 3, September 1967

pp. 496-502

Research Paper No. 324

of the

Division of Building Research

Price 10 cents

OTTAWA

July 1967

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1,'auteur a Blabore unc repr6sentation math6maticluc detaillee de la d6formntion

plastique, fond& sur la thBoric des d6formations de Dorn; cette repr6sentation est

utilisable pour le calcul du processus de d6fornlatio11

?L

des temperatures croissant

constammerit et, sous une charge variant lentement. I1 montre

clu'h

l'esception des

d6formations tertiaries, tous les elements de la courbe de d6formatiori des mat6riaux

de construction polycristallins sont d6terminCs par deux param&tres d6pendant

des contraintes,

et 2. L'auteur propose des fonctions explicites exprimant les

processus de d6formation. Ces fonctioils ne pcuvent pas &tre considBrBes rigoureuse-

ment parlant comme des Bquations d'Btat, mais ellcs dorirlerit dcs r6sultats accept-

clu

ables dans 1e:s calculs d'ingBnieurs, m&me si

- =

0. dt

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T. Z. H A R M A T H Y

A

Comprehensive Creep Model

Research Officer.

5

Fire Research section^ _{Based on Darn's creep theory, a comprehensive creep model has been developed that i s}

Division o f Building Research,

National Research Council, applicable to the calculation of deformation processes at steadily increasing temperatures O t t a w a , Ontario, C a n a d o and slowly varying load. I t i s shown tlzat the entire cozirse of the creep curve (excepting the tertiary creep) of str7~ctziml (polycrystalline) materials i s t~niqziely determined by two stress-dependent parameters, efo and

2.

_{Explicit expressioizs for the description of} creep processes are proposed. These expressio~zs cannot strictly be regarded a s equations of state, but yield aiz acctiracy ncceptable for engineering calculations even if d a / d t #

0. Introduction

where 0 is defined a s

THE author has often faced the problem of predicting the deformation history of steel structures a t rather rapidly increasing temperatures and occasionally a t slowly decreasing load, a s may occur in the exposure of structures t o fires in buildings. T o solve such problems a comprehensive creep model is needed t h a t is free of any restriction wit,h regard t o temperature and stress conditions, a t least within the domain of interest. From among widely accepted creep theories, t l u t propounded by Dorn [1,2, 31' has been selected t o form the backbone of a comprehensive model, not only for its own merit bnt also because of the convenience i t may offer in numerical creep analyses.

I n this paper any time-dependent deformation will be referred t o a s creep. As the anelastic strain is usually just a small fraction of the total creep strain, all time-dcpendent strains will be regarded a s essentially nonrecoverable; the term "creep strain," therefore, r i l l be used in place of "plastic creep strain" for con- veiiierice.

Only uniaxial creep will be considered, during the primary and secondary periods, and purely from a phenomenological poir~t of view. Thus no effort will be made t o interpret certain conclusions in the light of the dislocatiori theory.

Some Corollaries of Dorn's Creep Theory

T h e basic statement of Dorn's theory is as follows: I n deformation processes t h a t develop a t some co~lstailt stress level, the creep strain is a unique function of the stress and of a so-called temperature-compensated time, 8,

'

Numbers in brackets designate References at end of paper. For convenience, the d o l d t = 0 for t

>

0 restriction will be re- peated only in connection with concluding equations.

Contributed I>y the Metals Engineering 1)ivision and presented at the Metals Engineering Conference, Houston, Texas, April 3-5, 1967, of

THE

AMERICAN SOCIETY OF MECH,\NICAL ENGINEERS. Manuscript

received a t ASME I-Ieadquarters, January 30. 1967. Paper No. 67- Met-20.

For pure metals and dilute alloys the activation energy of creep,

AFI,

is inierlsitive t o (a) the microstructure of the material, (b) the temperature, if it is higher than one half of the melting temperature and the material is physico-chemically stable, and (c) the stress, unless i t is estrernely high.

AH

has been proved t o be approximately e q ~ i a l t o the activation energy of self diffusion.

Although tliere have been some doubts concerning the strict validity of combi~iing ten1peratiu.e and time in a single-state variable [4], the author found that with the steels so far studied, I)ornls theory proved a valuablc means of presenting creep d a t a in a concise form.

I n Fig. 1 3 family of e, versus 6'curves for an ASTM A36 steel

is shown a t several a = corist values. The curves represent some of the results of a comprehensive creep study still in progress in the author's laboratory [5]. J u s t a s with the conventional el versus 1 curves ( a t T = const and a = const) these curves also eshibit straight-line sections. T h e interval over which de,/dO is approsimately constant (and equal t o (del/d8),) will be referred t o here a s the secondary period of creep. Unless the temperature is constant, the creep rate in the secondary period of creep, (de,/dl),

=

E,,,

is not constant, so t h a t the espressions "secondary creep" and "constant-rate creep" are, as a rule, iiot interchange- able.

Since the shape of the el(@ curves is uniquely determined by a, ally feature of these curves is solely a function of a. Although this statement is quite straightforward, it h m far-reaching consequences, some of which have not been fully realized even by the promoters of the theory. These consequences will now be discussed in some detail.

T h e first corollary of Dorn's theory is immediately obvious: the slope of the el(@ ciirves during the secondary period of creep, (de,/d€':,, depends solely on u.

Uy differentiating equation (1) with respect t o t the following expression is obtained for the creep rate:

Nomenclature

a = constant, defined by equation (19), dimensionless

E = modulus of elasticity, lb/in.2

f

= function

AH

= activation energy of creep, Btu/lb mole

R

= gas constant, Btu/lb mole deg R

1 = time, hr

T = temperature, deg R

.I/ = variable defined by equation (IS), dimensioiiless

Z = Zener-Ilollomon parameter, hr-I

A

= increment

e = strain; without subscript: total load strain, dimensionless

i = strain rate, hr-I

0 = temperature-compensated t.ime, hr

,t

= variable defined by equation (23), dimensionless

a = effective creep stress, lb/in.2

5 = act;ual stress, lb/in.a

= function

cp = function

#

= function

c = constant

I

= limiting

0 = extrapolated t o the e, axis

s = in the secondary period of creep 1 = time-dependent

u

= ultimate, pertaining t o a,

(5)

Fig. 1 cr(6') curves

where the relation

has been utilized (from equation (2)). In general

but, as stated previously, during the secondary period of creep

Applying equation (3) to the secondary period of creep, (de,/de), may be expressed in terms of the secondary creep rate, it,, as follows:

There the symbol Z was introduced, since (dc,/dO), could be recognized as the well known Zener-Hollomon parameter [6] which, in agreement with the first corollary, had been proved to be a function of stress only long before publication of Dorn's theory. In this light, Dorn's theory should be viewed as an extension of the validity of equation (1) to the primary period of creep. By vir- tue of equation (6), the first corollary may thus be stated as3

du

Z

= Z ( u ) when - = 0 for t

>

0

dt (8)

The second corollary of Dorn's theory is that the intercepts, eta (Fig. I), obtained by extending the straight-line sections of the el(@ curves to the el &xis are also uniquely determined by stress alone and are independent of temperature. Thus

d a

€10 = eto(a) when - = 0 for t

>

0

dt (9)

obtained in the earliest stages of the creep tests are often rather i n a ~ c u r a t e , ~ and (c) is, in general, a poorly reproducible quantity.

An indirect proof may, however, be provicled by carrying out creep tests a t some constant stress but varying temperature.= If after plotting E, (referred to any arbitrarily selected E , = 0

level) against 19 it is found that in a reasonably wide range of 0 the points fall on a straight line, and the slope of this line is in good agreement with the value of the Zener-I-Iollomon parameter a t the employed stress, then the validity of equation (9) is proved.

Fig. 2 summarizes the results of such a creep test performed on an A36 steel. I t may be seen that the ~ ~ ( 0 ) cuive, in spite of the marked variation of the temperature, does contain a straight-line section. With the aid of conventional creep records, this section could be identified as representing the secondary creep process. Fig. 2(b) also shows that the slope of this section is in fair ngree- ment with the value of Z a t the applied stress (7500 psi). There is good reason to believe, therefore, that equatio~i (9) is a t least a

fair approximation and is acceptable for engineering calculations. Even if equation (9) were grossly inaccurate, there wo~dd still be a strong temptation to use it because of the remarkable convenience it offers. This will be more fully understood after dis- cussion of the third corollary.

I t may be of interest to note that for A36 steel, which nrns used in the experiments referred to in connect,ion with Figs. 1 and 2,

A H / R was found to be 70,000 deg It. Furthermore,

0 . 0 2 6 ~ ' . ~ if a

_<

15,000 z = (

1.23 X 1 0 L ~ 0 ~ 0 0 0 3 w for 15,000

<

u

<

45,000 and

= 1.7 X 10-lo

Equation (11) has been derived from a plot of badly scattered points. There are some indications that the stress esponei~t should probably lie between 2 and 3.

Although the applicability of Dorn's theory is generally restricted to temperatures above one half of the melting temperature, for carbon steels the above formulas seemed to be applicable within the 750 to 1300 deg F inteival, which actually represer~ts This conclusion seems to be a t variance with McVettyls find-

the entire temperature domain of practical

ings [71. I t is very difficult to furnish direct proof of the validity

In many applications it has been elltirely satisfactory to model

of equation (9). The difliculties are associated with the facts

the creep curves according to suggestioll of Odqvist,s [S, 9l as bhat: (a) a t t + 0,

i ,

+ m, so that the elastic and creep strains

-

cannot be separated unambiguously; ( b ) the deformation records With differential transformer type extensometers the strain

-

record is falsified by the bending and uneven heating of the estension

This first corollary was discussed in Dorn's first paper, and is arms during the initial period of fast deformation.

derived here in a somewhat different way for the sake of complete- 6 The result of a siinilsr test was reported by IDorn [ I ] , to prove the

ness. validity of equation (1).

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Fig. 2 Results of an experiment concerning the eto(a) relation; (a) temperature and strain history of the specimen (a = 7500 Ib/in.2); ( b ) ~ t ( 6 ' ) plot

consisting of a qllasi-instantrtt~eous section (line OA in Fig. 1) followed immediately by the section of secondary creep (line AB). Thus Z and arc the only creep parameters of interest and the two corollaries are already capable of summarizing the "creep properties" of the material in a concise form. I n the third corollary of Dorn's theory it will be shown, however, that even more realistic creep curves can be produced with the aid of these two stress-dependent parameters.

Division of equation (5) by equation (8) gives

By eliminatilig, first

B

with the aid of equtttion ( I ) , then a with the aid of equation (9), one obtains

The following conditions must obviously be satisfied:

( a ) At

B

= 0 (or t = 0) el = 0, and since de,/dB (or det/dt) is

known to be extremely high, according to equation (12)) @

=

0. Also, according to equation (14),

,

= 0.

( b ) As 6' + m (or t -P m) el -f m 6 and deL/d6' -t Z (i.e.

secondary creep prevails), so that @ -P 1. Also, as the curves

in Fig. 1 indicate, E , -t

+

ZB; consequently

,

--t (el - €10).

T o summarize:

which can also be written as

Integration gives

when

B

t m , r, t m and

{it

( 1 6 ~ )

+ (61

-

€ t o ) (16b)

I t is well known, furthermore, that de,/dB is a decreasing mono-

-

Here the fact that at some high value of er tertiary creep will set off is not taken into account.

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tonic function of el; Qj and I) must therefore be increasing mono-

tonic functions of E,.

From equations (14) and (IGb) i t follows that

Clearly, since elo is determined by the ~B(E,) relation, it cannot be regarded as a truly independent variable in equation (12). Con- versely, if efo is considered a s the basic irlformation, arrived a t ex- perimentally (and presented by equation (9)), equation (17) shollld be regarded as a condition that restricts the selection of the @(el) f u ~ ~ c t i o ~ l . T O relieve this restriction, E , and may be combined in a single variable, a s

The solution of this equation is

By substituting e q ~ ~ a t i o n s (8) and (0) ill this eql~ation an explicit form of equation ( I ) is obtained.

T h e creep curves yielded by equation (27) run, generally, well below the experimental creep curves. A ml~cln better agreerner~t may be obtained by selecting y = tnnh2 11. For this fllnction a = 1, so that

I n this way equation (17) becomes Wilh the aid of equation (24) (I!))

Thus any ~ ( y ) function, which is monot,onic increasing t h r o ~ ~ g h - out the domain y

>

0 and for which the conditions

are fnltilled and the improper integral given by equation (19) exists, can he selected a s the function on the right side of equation (12). Thus eqt~alion (12) may be rewritsten a s

1 de, ... _-- - I _when d a _--₌₀ Z (18 y(ae /E!o) dl

Although the foregoing a r g ~ ~ r n e ~ r t , s do not imply t l ~ n t the same cpk) function is applicable to any stress lcvel, there is good reason to believe (to be touched upon later) that cp(;~j) is practically independent of stress. Even further, i t seems probable t h a t the same cp(y) fiunction mag be used for larger groups of materials.

As the solution of equation (21) does rrot, as a rule, orfer a n explicit expression for E,, it is convenient to present it,s solution in the form

cla

(=((%8)

for - 0

E 10 rll

wl~ere is a new variable d e f i ~ ~ e d as

( is, in fact, a quantity proportional t o the primary creep strain. Thus equation (22) represents the primary creep versus temperature-compensated time relation in a dimensionless form.

An expression for Z O / E ~ ~ is obtair~ed by integration of equation (21):

ze

E l

- - _-..- _-txrh(:) _{for ;;=(I}(la

E l 0 El0

(29)

and of eq11ai.ion (25)

Retaining only the first two terms in the Taylor's series of the hyperbolic tangent function, E ~ / E , ~ may be readily expressed from equation (29),

or taking tanh ?j = y and connl~ining eql~at~ions (23), (30), ant1 (31 ), one obtains

This e q ~ ~ a t i o n proved a fair approsinlation of thc creep curve within the 0

<

el/elo

<

0.5 i n t e ~ v a l .

Equation (32) may be recognized as an elaborate form of the celebrated Andrade's equation (in terms of 0 instead of t ) , which was proposed on experimental grolunds over 56 years ago [ l o ] and defended recently [ I l l . T h e fact that the primary creep is indeed proportional to the power of t (or 0) for a variety of materials, irrespective of the applied stress, seems t o support (a) the selection of the fl~nction y = tanh2 y, and (b) the a s s ~ ~ n ~ p t i o n that this function map be of fairly wide application.

T o facilitate creep deformation calculations, two ( versus ZO/elo plots, corresponding t o the foregoing two selections of the cp(y) function, are presented in Fig. 3. I n general, curve b is expected to give higher accumcy. This curve and equations ( I ) and (11) were used to plot the Litheoretical" creep curves in Fig. I. From the fact that the time appears in the di~nensior~less group Z O / E ~ ~ one may conclude that the higher the value of the secondary crecp rate, the faster is the npproach to thc conditions of secondary creep. This finding is ill full agreement with obse~vations made by Garofalo [12] and recently a t the Division of Building Re-

search [5]. Thus, making use of equation (23)

d u

( =

'

-

L ' " \ ( a

z)

d )0:( wlien - = 0 (25)

The Mechanical Equation of State

E 10 dl Without the da/dt = 0 r e s t r i c t i o ~ ~ equation (21) could be re-

garded as a mechanical equation of state. It is of considerable in- I n a number of calculations the author selecled the fur~ction terest to see whether i t is possible to lift the da/dt restriction, and

cp = tanh y, mai111y because with this s e l e c t i o ~ ~ all explicit espres- if so, under what conditions.

sion for E, can be obtained. For this function the value of the im- T h e possibility of the existence of mecha~~ical equatior~s of

proper integral (see equation (10)) is ln2, so t h a t equation (21) state has been discussed by many authors (see, for example, a becomes review presented by Lubahn [13]). It is generally believed t h a t

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ze/&to

Fig. 3 Dimensionless plots of creep curves; (a) (o = tan h [In 2(61/61")]; (6) (o = tan h2 (dl/dlO)

such equations may be applicable to certain very restricted 2, = i C e A " / R T (33) situations. Dorn definitely opposed the view that mechanical

equations of state may be applicable to creep. For example, for A36 steel, using equatioll (Ib) and AHlR = I t is usual to test the validity of equations of state by com- 70,000 deg It, the following expression results:

paring stress-stmin cul-ves (for which de,/dt

=

&

=

const) derived from creep illformation with experimental stress-strain curves. Although a satisfactory agreemeut would, beyond doubt, prove the unlimited applicability of certain equations of state, it must be pointed out that from a practical point of view this kind of test is unduly severe. I n the t,ypes of problem with which the author is concerned 10,000 psi/hr can be regarded as an adverse value of da/dt, but during conventional tensile tests da/dl may be several orders of magnitude higher than this value.

The creep i~~formatior~ and the unrestricted form of equation (21) (or of its explicit form presented by equation (28)) proved unsatisfactory beyond doubt for the calculation of the initial por- tion of the stressstrain curves (at E, = const) of A36 steel. The calculated strains are always very ~ n u c h lower, sometimes by a factor of 100, than the strains pertaining to the same stress history in experimental stress-strah~ relations. Nevertheless, as duldl decreases, the calculated strain rate, de,/dl, seems to a p proach

&.

Every stress-strain curve exhibits a section in the vicinity of

a, for which da/dt = 0 (where both a, and a should be referred to the original cross sectional area). During that brief period that du/dl ..:0 and a

=

a,, the conditions are similar to those

during an early stage of the tertiary creep of conventional creep tests, so that it may be possible to express a, in terms of the strain rate

&

a t which the tensile test is carried out. Assuming that in the vicinity of a, the process of deformation corresponds t o that during secondary creep, one may attempt t,o calculate a, with the aid of the Z ( a ) relation for the material examined as the stress pertaining to Z,, where, in virtue of equation (7)

70,000

A=(-

+

_{In€, for 15,000}

_<

_a

_<

_45,000 3333 I'

I t may be of interest to note that, on experimental grounds, Ludwik [14] found a similar expressio~~ for the strain-rate dependence of stresses in tensile tests almost five decades ago, and that his observations have since been coufirmed [15].

In Table 1 some values of a,, calculated for an A36 steel with the aid of equation (33), and of an appropriate Z ( a ) plot (the

a

5

45,000 psi range of which has been approximated by equation (I)) are compared with those obtail~ed by tensile tests carried out a t €, = 0.075 in./itl. min strain rate. Although the experimental values are consistently lower than the calculated ones, the discrepancies are not excessive and may be explained

Table 1 Comparison of measured and calculated ultimate stresses

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by the inadequacy of the assumptions used. These findings indicate, however, that in advanced stages of plastic deformation, i.e., when €1

>>

E,O, the rate of deformation is relatively insensitive

to the previous stress history of the material and that it corresponds, a t least approximately, t o the secondary creep rate a t the prevailing (constant) stress and temperature.

As the rate of the deformation is controlled primarily by Eta

during very early stages of plastic deformation, and primarily by

Z when el

>>

€10, i t may be concluded that of the two stress- dependent parameters is much more sensitive than Z to the manner of applying load. I n arriving a t such a conclusion one must presuppose that the lifting of the du/dt = 0 restriction from equation (21) may be compensated for simplv by regarding

€10 and Z as quantities dependent on the stress history in addition

t o the momentary values of the stress, without changing the form of the equation. There are, however, some doubts that such an assumption is applicable when u varies very rapidly with time.

T o investigate the applicability of equation (21) in general, or of equation (28) in particular, to cases where a varies slowlv with time, a creep test was carried out on an A36 steel specimen a t 1000 deg F, with a varying stepwise in small increments (Fig. 4). I n the same figure the experimental and calcrilated ~ , ( t ) curves and points indicating the average strain rate, Ae,/At withiii periods of constancy of a , are also shown. The calculations were performed with the aid of the following finite differei~ce form of equation (28)

(and of equations (I) and (11)), except for the first time interval, in which the deformation was calculated with the aid of curve b of Fig. 3. The experimental curve indicated a stronger dependence of Z on u than that expressed by equation (I). Nevertheless, the discrepancies between the experimental and calculated values are no higher than one would normally expect in creep tests. I t may be assumed, therefore, that the validity of equation (21) (or that of equation (28)) may be extended t o situations in which the stress is a slowly varying function of time.

S o m e Complementary Assumptions

The creep model so far evolved is applicable strictly to deformation processes occurrir~g a t temperatures above one half of the meltr ing temperature of the material.' Fortunately, the temperature limitation rarely presents any practical difficulties. Eve11 though the mechanism of plastic deformation a t low temperatures may be entirely dirferent from that a t higher temper:~tures, from a phenomenological point of visw there is no need to tli~criss the creep a t low and high temperatr~res separntely. I'ril~~ary creep does exist a t any t e m p e r a t ~ ~ r e , if the stress is high enough. The observation that a t sutficie~ltly low temperatures tle,/dt -+ 0 as

t -+ m does not conflict, a t least in a qualitative sense, with

Dorn's theory. For an A36 steel, for example, the application of equations ( I ) and (7) would yield a room temperature secor~dary creep of about G.4 X in./in. hr a t 40,000 psi. Obviously, there is no practical difference in sayirlg that a t room tenlperatr~re carbon steels do not exhibit secondary creep, or that their secondary creep rate is far below aliv detectable level. Thus, although the errors associated wiLh the unauthorized extension of the creep modcl to lower temperatures may be very sigilificant on a percentage basis, they remain insignificant as far as the total strain is concer~led.

It

appears, therefore, that the following assumption is justifiable in many engineering problems:

Assumption 1. When studying the deformation history of metal

structures a t steadily increasing temperatures, i t is permissible to 'As mentioned earlier in connection with formulas (I) and (11) it is probable that for many materials the strict applicability of the model may be extended to significantly lower temperatures.

extend the validity of the previously described creep model t o lower temperat~lre regimes.

By accepting Assumption 1 i t is implied that plastic deformation (at least primary creep) always develops following the elastic deformation of the material. I t is well ltnown, however, that a t lower temperatures, in general, only elastic deformation will occur if the stress is lon~er than some limiting value, u,, which is a func- tion of the temperature. With carbon steels, UI may be taken as the stress a t the true yield point. T o eliminate this difficulty, a further assumption may be introduced:

Assumption 2. In the previously described creep model, the

stress, u, sho111d be interpreted as the "effective creep stress," which is related to the actual stress, 5, in the following way:

where

At suficiently high temperatures a, = 0, so that there is no need to differentiate between effective creep strcss and actual s t r e s ~ . ~

The total (uniaxial) deformation of a metal element can thus be detailed by the following equations:

According to the conclusion reached in connection with Fig. 4, equation (37a) may be applicable not only when cla/dt = 0 but also when the strcss varies slowly with time.

I n Fig. 5 the cnlculated variation of the central deflectiorl during fire exposure of a beam assembly (consisting of a W F 12 by 6l/2 steel beam, coiicrete deck and sprayed insulation) is compared wit,h an experimental deflection curve. The calculations have been carried out with the aid of equations (26) and (37a); details of the calculations have been given [16] ; arid the experiment has been reported 1171. The agreement between the es- perimental and calculated curves is satisfactory from ail engineer i ~ i g point of view.

References

1 Dorn, J. E., "Some Fundaniental Esperitnents on High Tern- perature Creep," Journal o f ~Mecl~anical Pllus Solids, Vol. 3, 1954, p.

85.

2 Dorn, J. E., "Progress in Understanding High-Temperature Cleep," Gillett lMemorial Lecture, ASTM, 1962.

3 Dorn, J. E., and mote, J. D., "Physical Aspects of Creep,"

Higll Ten~perature Structures and Materials, edited by Freudeothal,

A. .M., Boley, B. A., and Liebowitz, II., MacA'Iillan Co., New Yorlc,

1964, p. 95.

4 Garofalo, F., Smith, G. V., and Boyle, B. W., "Validity of Time-Compensated Temperature Parameters for Correlating Creep and Creep-Rupture Data," ASlME I'aper No. 55-8-164, 1955.

5 Harmathy, T . Z., and Stanzak, W. W., "The Creep Properties of Structural Steels." to be published.

6 Zener, C., and Hollolnon, J. H., "Effect of Strain Itate 011 the

Plastic Flow of Steel," Jourrlal of Appl. Phys., Val. 15, 1944, p. 22.

7 McTretty, P. G., IVorking Stresses for EIighTernperature Service," ~Mech. Eng., Vol. 56, 1934, p. 149.

8 Odqvist, F. I<. G., "Engineering Theories of Metallic Creep," Symposium su la plastieita rlelle scienza delle construrioni, Varellna,

1956.

9 Odqvist, F. I<. G., and Hult, J., I<riec/lfistigkeit n~etallischer Werl;stoffc, Spriilger-Verlag, Berlin, 1962, p. 168.

8 Deformation processes developing at moderate stress levels can

generally be studied without making use of Assumptions 1 and 2.

In fact, in his calculntions the author never used these nssumptions. They might become essential, however, if for some reason, e.g.. re- straint against thermal expansion, there is any probability that excessive stresses will develop.

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Fig. 4 Creep at variable stress

C x y e r ~ m e n t a l

- - - C a l c u l a t e d

Fig. 5 Central deflection of a beam during fire exposure

10 dn C. Anclrade, E. N.. "The Viscous Flow in Metals and Allied 14 Ludwik, P.. Ele7nente d e ~ t e c h ~ ~ o l o g i s c h e ~ ~ dIechanik, Springer- Plienomelia," Proceedi~~gs of the Royal Societu, Vol. A84, 1910, p. 1. Verlag, Berlin, 1909, p. 44.

11 da C. Andmcle, E . N., "The Validity of the t l / 3 Lam of Flow of 15 Nadai, X., Theorg of Flow and Fractzcre of Solids, Vol. I , 2nd

Metals," Phil. lllag., Vol. 7, 1962, p. 2003. ed., McGraxv-Hill, New York, 1950, p. 21.

12 Garofalo, l;., "Resistn~ioe to Creep Deformation and Fracture 16 I-lar~natl~y, T . Z., "Deflection and Failure of Steel-Supported in Metals nncl Alloys," ASTM Spec. Tech. Publ. No. 283, 1960, p. 82. Floors and Be:~ms in Fire," ASTM Speci:rl Teclir~icnl Publication,

13 Lubalin, J. D., "Creep Properties and Creep-Stress-Time No. 422, 1967.

lielations," i n P ~ o c e e d i r ~ g s f o ~ a S h o ~ t Cocwse, dIechanics of Creep, 17 Bletznclter, R. \IT., "Effect oi Stmctur:ll 1iestr;iint on the Fire edited hy J. Marin, Dept. Eng. hIech., Pennsylvnnia State Uni- Resistance of Steel Buildiiig Fmmcs," A S T h I Specid Teclinical

versity, 1954. Publication, No. 422, 1967.

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